The character and personality of Gandalf is, I feel, fairly consistent between film and book. If we look at Ian McKellen’s words about how he saw Gandalf, and if we remember that he is a huge fan of the books, it is easy to see why the two coincide:

“To act, I preferred Gandalf the Grey. He’s more complicated than Gandalf the White. He had enormous strength, resilience, intelligence and determination, passion and generosity. He was also very human, very frail, in the sense that he liked to drink, he liked to smoke, he liked to laugh, he liked to play. He also was human in the sense that he was worried he wasn’t doing the job properly–that he’d somehow let Middle-earth down by not anticipating Sauron’s revival. He had to really organize himself. That was a fascinating character to play.” (www.lordoftherings.net)

What is different is the amount that Gandalf appears through FotR, and the amount he shows his Maiar powers.

Enter Gandalf!
In the films, Gandalf is first seen on the road to Hobbiton, singing “The Road Goes Ever On”, the back of his cart laden with fireworks. He meets Frodo on the road, greeting him warmly and saying his now famous line: “A wizard is never late, Frodo Baggins. Nor is he early. He arrives precisely when he means to.” Frodo then talks to Gandalf as the cart trundles into Hobbiton, both about the long-expected birthday party and Bilbo’s recent odd behaviour.

In the book, he also comes to the Shire in a cart, but he does not meet with Frodo.
“He wore a tall pointed blue hat, a long grey cloak, and a silver scarf. He had a long white beard and bushy eyebrows that stuck out beyond the brim of his hat.” (FotR, A Long-Expected Party)
A toned down version of this description clearly fits Ian McKellen’s Gandalf – with the omission of the scarf and the eyebrows reaching beyond of the brim of the hat!

In both the film and the book, a group of hobbit children soon gather round the cart, hoping that it Gandalf has brought fireworks with him. In the film, the children got a sneak preview of the fireworks, in the book, they didn’t.

“Hobbit children: Gandalf! Gandalf!
(Some hobbit children run down the grassy hills towards the carriage)
Hobbit children: Fireworks, Gandalf! Gandalf! Fireworks, Gandalf!
(They look at him in disappointment as he drives along, pretending not to notice them.)
Hobbit children: Awwwww.
(Suddenly, a blast of fireworks goes off from the carriage. The hobbit children clap their hands and cheer.)”

Gandalf in the film is simply introduced as a harmless old man, a friend of the children of Hobbiton, a magician whose stock in trade was fireworks and tricks. There is slightly more to him from the start in the book:
“the old man was Gandalf the Wizard, whose fame in the Shire was due mainly to his skill with fires, smokes, and lights. His real business was far more difficult and dangerous, but the Shire-folk knew nothing about it.” (FotR, A Long-Expected Party)

In the book the essential dichotomy of Gandalf is shown right from the start – that he is unassuming and humble, a friend to the hobbits – but that he is also a true power in the world, knowing of the Dark Lord’s plans and expected to be able to counter them.

The Party
The fireworks display is similar in the book and film, with the finale being a swooping red-golden dragon. But the dragon firework was not let off by Merry and Pippin, and Gandalf didn’t catch them doing so. In the book, Gandalf knew what Bilbo would do with the Ring, providing a flash of light to make the hobbit’s appearance look less remarkable. In the film, he looked distinctly surprised as Bilbo vanished.

Bag End
The scene at Bag End, where Bilbo almost refuses to give up the Ring, is similar in both film and book. Even the part where Gandalf seemed to grow tall and full of power is described in the film as per the book “he seemed to grow tall and menacing; his shadow filled the little room.”

After that scene, Gandalf disappears almost immediately in the film, just after Frodo finds the Ring on the floor. The wizard leaves suddenly – imploring Frodo to keep the Ring secret and safe – and rides to the Library in Minas Tirith, where he reads Isildur’s scrolls, and discovers the fact that only fire can show the writing on the One Ring. (In the book, the section about Isildur’s scroll is told during the Council of Elrond).

In the book, however, Gandalf directs Frodo to an envelope on the mantelpiece, which contains the Ring. Gandalf then tells the young hobbit that Bilbo wanted him to now own Bag End – and again implores Frodo to keep the Ring secret and safe. The next day, Gandalf and Frodo sit down and talk for a long time, Gandalf saying that he had had suspicions about the nature of the Ring.

There is also a clear sense that not all the people of the Shire consider Gandalf to have good and pure intentions: “I’ll come and see you again, as soon as I can. Expect me when you see me! I shall slip in quietly. I shan’t often be visiting the Shire openly again. I find that I have become rather unpopular. They say I am a nuisance and a disturber of the peace. Some people are actually accusing me of spiriting Bilbo away, or worse. If you want to know, there is supposed to be a plot between you and me to get hold of his wealth.” (FotR, A Long-Expected Party)

“[Bilbo] had undoubtedly fallen into a pool or a river and come to a tragic, but hardly an untimely, end. The blame was mostly laid on Gandalf.” (FotR, The Shadow of the Past)

The words “a disturber of the peace” are taken out of context in the film, used as a joke, rather the serious matter of the book. Indeed, in these scenes, Gandalf seems care-worn, tired:

“Frodo thought the old wizard looked unusually bent, almost as if he was carrying a great weight.” (FotR, A Long-Expected Party)

Time passes
Where the book and film first differ significantly is the time scale involved between the discovery of the Ring and the onset of Frodo’s journey. In the film, there is no indication of the huge lapse in time between Bilbo’s party and the start of the journey to Rivendell. In the book, Gandalf was initially gone from the Shire for three years, then turned up ever now and again for the next year or so. After that, Frodo did not see him for over nine years, and had almost given up thinking that the wizard still cared for hobbits before he once again reappeared.

The One Ring revealed
In the film, Frodo had put the Ring in its envelope away in a chest, and Gandalf revealed it as the One Ring by throwing the envelope into the fire. The envelope burned away leaving the glowing Ring. In the book, Frodo had always kept the Ring in his breeches pocket on a chain that hung from his belt. He removed it from the chain and gave it to Gandalf, who again revealed the writing by throwing it into the fire.

In both versions, Gandalf then gave Frodo a potted history of the Ring – though with a great deal more detail in the book. He told Frodo of Gollum, that the creature knew of the Ring’s location, and also that the information was likely to have been tortured out of him – so that the Shire was no longer safe for the Ring or the hobbit. He concludes by telling Frodo that the Ring was meant for him alone.

In both book and film, Frodo then offers the Ring to Gandalf.

Film – “Gandalf: (yelling) Don’t tempt me Frodo! (calmly continuing as Frodo withdraws his hand) I dare not take it. Not even to keep it safe. Understand, Frodo, I would use this Ring from a desire to do good, but through me, it would wield a power to great and terrible to imagine.”

Book – “No!” cried Gandalf, springing to his feet. “With that power I should have power too great and terrible. And over me the Ring would gain a power still greater and more deadly.” His eyes flashed and his face was lit as by a fire within. “Do not tempt me! For I do not wish to become like the Dark Lord himself. Yet the way of the Ring to my heart is by pity, pity for weakness and the desire of strength to do good. Do not tempt me! I dare not take it, not even to keep it safe, unused. The wish to wield it would be too great for my strength. I shall have such need of it. Great perils lie before me.” (FotR, The Shadow of the Past)

While the film suggests that the Ring would start to drive Gandalf’s actions if he bore it, the book truly shows the struggle that Gandalf would undergo if he bore the Ring. Galadriel’s transformation speech in Lothlórien seems obviously adapted from Gandalf’s book speech.

In both versions, Gandalf then catches Sam listening outside of the window, and enrols him on Frodo’s journey. In the book, Gandalf then stayed in the Shire for another two months, before he again leaves suddenly, promising to be back in time for Frodo’s birthday party – which had also been set as the leaving day for his journey. But he didn’t appear …

Many Meetings
The next time Gandalf appears in the book is at Rivendell. In the film, Gandalf’s next appearance is a flashback one of him and Saruman in Isengard, but in the book this tale is recounted during the Council of Elrond.

In both book and film, he is in Frodo’s room when the hobbit first awakes, and tells Frodo that he is now safe in the House of Elrond.

A change in persona is seen in the book during the feast in the Hall of Rivendell. Gandalf the Istar is truly revealed to Frodo for the first time:
“Frodo looked at them in wonder, for he had never before seen Elrond, of whom so many tales spoke; and as they sat upon his right hand and his left, Glorfindel, and even Gandalf, whom he thought he knew so well, were revealed as lords of dignity and power.
…
Gandalf was shorter in stature than the other two; but his long white hair, his sweeping silver beard, and his broad shoulders, made him look like some wise king of ancient legend. In his aged face under great snowy brows his dark eyes were set like coals that could leap suddenly into fire.” (FotR, Many Meetings)

The Council of Elrond
In the book, Gandalf and Bilbo talk in the gardens before the Council, while in the film, action starts straight in at the Council.

The wizard’s long account of his wanderings in the book starts when he went to Dol Guldur to seek the Necromancer, and continues through his entire search for Gollum. He talks of the letter he sent to Frodo via Barliman Butterbur (not in the film), and then of his visit to Isengard. Much of this is missing in the film, with emphasis placed firmly on his time with Saruman. And even there, while the basics of the wizards’ conversation are the same, the details are different, as shown below:

– Book – Radagast tells Gandalf of the Nine riding forth, Film – Saruman tells Gandalf
– Film – there is no discussion of Saruman the White becoming Saruman the Many Coloured
– Book – there is no wizard’s duel and Saruman does not consider that Gandalf chooses the “way of pain”
– Book – “they” (presumably Saruman’s servants) took Gandalf and placed him on the pinnacle, Film – Gandalf was shot up there through Saruman’s magic.
– Book – Gwaihir – already known as King of Eagles – came unlooked-for, Film: Gwaihir was called by Gandalf’s magical moth
– Book – Gwaihir sets him down in Rohan, Film: set down point unknown

The absolute importance of Gandalf’s tale was made much more clear in the book:
“And that, Frodo, is the end of my account. May Elrond and the others forgive the length of it. But such a thing has not happened before, that Gandalf broke tryst and did not come when he promised. An account to the Ring-bearer of so strange an event was required, I think.” (FotR, The Council of Elrond)

There are also a number of other differences between Gandalf’s part in the Council in the film and the book.
Along with the cutting of Tom Bombadil in the film, Gandalf saying that the Ring could not be given to him was omitted. In the film, Gandalf adds himself to the Quest during the Council, while in the books, he only decides to join the Fellowship some time after the Council: “I do not know if I can do anything to help you; but I will whisper this in your ears. Someone said that intelligence would be needed in the party. He was right. I think I shall come with you.
So great was Frodo’s delight at this announcement that Gandalf left the window-sill, where he had been sitting, and took off his hat and bowed. “I only said I think I shall come. Do not count on anything yet.” (FotR, The Ring Goes South)

Caradhras
On Caradhras – in the book, Gandalf uses his magic obviously for one of the few times in the whole of “The Lord of the Rings”.

“Picking up a faggot he held it aloft for a moment, and then with a word of command, naur an edraith ammen! he thrust the end of his staff into the midst of [the wood]. At once a great spout of green and blue flame sprang out, and the wood flared and sputtered.”

There are a number of other differences on Caradhras, including Gandalf handing round a flask of miruvor in the cold of the evening in the book but not the film. In the film, Gimli brings the subject of Moria up, insisting again and again that they should go through the Mines. However, in the book, it was Gandalf who first brought up the subject: “But there is another way, and not by the pass of Caradhras: the dark and secret way that we [he and Aragorn] have spoken of.” (FotR, The Ring Goes South)

The ultimate decision to abandon Caradhras was also made by Gandalf in the book, but by Frodo in the film.
“Gandalf: Let the Ring bearer decide.
Boromir: We cannot stay here! This will be the death of the Hobbits.
Gandalf: Frodo?
Frodo: We will go through the mines.
Gandalf: So be it.”

The wargs
A small scene appears in the book, but not the film, where the Fellowship are attacked by wargs on the way from Caradhras to Moria. Again Gandalf shows his true strength:
“Gandalf stood up and strode forward, holding his staff aloft. “Listen, Hound of Sauron!” he cried. “Gandalf is here. Fly, if you value your foul skin! I will shrivel you from tail to snout, if you come within this ring.””
“In the wavering firelight Gandalf seemed suddenly to grow: he rose up, a great menacing shape like the monument of some ancient king of stone set upon a hill. Stooping like a cloud, he lifted a burning branch and strode to meet the wolves. They gave back before him. High in the air he tossed the blazing brand. It flared with a sudden white radiance like lightning; and his voice rolled like thunder.
Naur an edraith ammen! Naur dan i ngaurhoth!” he cried.
There was a roar and a crackle, and the tree above him burst into a leaf and bloom of blinding flame.” (FotR, A Journey in the Dark)

Bill the Pony
In the film, it was Aragorn who told Sam to let Bill go while Gandalf was working on the door. In the book, it was Gandalf who told Sam to let the pony go:
“He laid his hand on the pony’s head, and spoke in a low voice. “Go with words of guard and guiding on you,” he said. “You are a wise beast, and have learned much in Rivendell. Make your ways to places where you can find grass, and so come in time to Elrond’s house, or wherever you wish to go.” (FotR, A Journey in the Dark)

The way into Moria
At the door into Moria, Gandalf readily admitted that he did not know the way to open the door in the book. But he said that he could think of ten score spells in the tongues of Elves and Men and Orcs that were used to open doors, which he promptly started working his way through. The ones actually mentioned in the book are “Annon edhellen, edro hi ammen! Fennas nogothrim, lasto beth lammen!” and “Edro, edro!”.

In the film, however:
“Gandalf: Oh it’s quite simple. If you are a friend you speak the password and the doors will open.
Gandalf: Annon edhellen edro hi ammen! *Gate of the Elves open now for me!* Fennas nogothrim lasto beth lammen! *Doorway of the Dwarf-folk listen to the word of my tongue!*
Pippin: Nothing’s happening.
Gandalf: I once knew every spell in all the tongues of the Elves, Men, and Orcs.
Pippin: What are you going to do, then?
Gandalf: Knock your head against these doors, Peregrin Took! And if that does not shatter them and I am allowed a little peace from foolish questions I will try to find the opening words.
Gandalf: Ando Eldarinwa a lasta quettanya, Fenda Casarinwa! *Gate of Elves listen to my word, Threshold of Dwarves!”

In the book, it is Gandalf who discovers the answer to the riddle:
“With a suddenness that startled them all the wizard sprang to his feet. He was laughing! “I have it!” he cried. “Of course, of course! Absurdly simple, like most riddles when you see the answer!” (FotR, A Journey in the Dark)
In the film, it is, of course, Frodo.

In Moria
In both the film and the book, it is Gandalf who guides the company through the dark of Moria, knowing where he wants to take the company, never faltering until he gets to an archway, containing three passages, that he doesn’t recognise.

The wizard’s anxiety about the journey through the mines is not so obviously shown in the film, whereas in the book it is clear:

“Gandalf was awake, though lying still and silent. He was deep in thought, trying to recall every memory of his former journey in the Mines, and considering anxiously the next course that he should take; a false turn now might be disastrous.
…
The last thing that Pippin saw, as sleep took him, was a dark glimpse of the old wizard huddled on the floor, shielding a glowing chip in his gnarled hands between his knees. The flicker for a moment showed his sharp nose, and the puff of smoke.” (FotR, A Journey in the Dark)

Chamber of Mazarbul
In the book, Gandalf stays behind in the Chamber of Mazarbul to hold the door while the Fellowship run on to the Bridge. While he’s there, he gives another display of his power. Frodo hears him muttering words, and then:
“there was a stab of white light. Then there was a dull rumble and a heavy thud. The drum-beats broke out wildly: doom-boom, doom-boom, and then stopped. Gandalf came flying down the slope and fell to the ground in the midst of the Company.
“Well, well! That’s over!” said the wizard struggling to his feet. “I have done all that I could. But I have met my match, and have nearly been destroyed. But don’t stand here! Go on! You will have to do without light for a while: I am rather shaken. Go on! Go on!”

“I found myself suddenly faced by something that I have not met before. I could think of nothing to do but to try and put a shutting-spell on the door … What it was I cannot guess, but I have never felt such a challenge. The counter-spell was terrible. It nearly broke me. For an instant the door left my control and began to open! I had to speak a word of Command. That proved too great a strain. The door burst in pieces.” (FotR, The Bridge of Khazad-dûm)

In the film, however, Gandalf knows what foe he faces.
“Boromir: What is this new devilry?
Gandalf: A Balrog. A demon of the ancient world. This foe is beyond any of you. Run! Quickly!”

On the way to the Bridge, he continues to try and work out what the creature was. An orc arrow lodged in his hat at the start of the bridge. At the Bridge, the Balrog appears and Legolas identifies it.
“”A Balrog!” muttered Gandalf. “Now I understand.” He faltered and leaned heavily on his staff. “What an evil fortune! And I am already weary.”” (FotR, The Bridge of Khazad-dûm)

The fight scene itself is then almost identical in film and book.

Conclusions
Gandalf is very similar in the book and the film, but what is really missing from the film are the extremes of his personality. While he’s seen as a friend to hobbits, a harmless old man making fireworks, the extent of his homeliness is never shown as well as in this single quote from the book:

“For myself I should like a pipe to smoke in comfort, and warmer feet.” (FotR, The Ring Goes South)

On the other hand, his Maia majesty and power is also not really shown in the film, presumably so that Gandalf the White looks that extra bit better against Gandalf the Grey. Also in the book is a good number of fire references, and while Gandalf is not revealed as the bearer of Narya until later in the book, a connection with fire is established much earlier. This is unfortunately missing in the films.