How did it happen in the movie?
The Fellowship has to travel very far – we get a whole lot of shots of the companions travelling wide-open spaces or climbing hill slopes.
During a rest on a hilltop the travellers get a chance to rest a little and plot the rest of their journey. While Boromir tries to teach the youngest Hobbits some sword-fighting, Sam cooks the Fellowship dinner and Gimli tries to persuade Gandalf to take the route under the Mountain, but Gandalf dismisses this idea.
The rest is rudely interrupted when Legolas spots a flock of crebain, birds in the service of Saruman, on the horizon; and the entire Fellowship has to seek cover.
Now that it has become clear that the passage south is being watched, Gandalf leads the company through the pass of Caradhras; but Frodo drops the Ring and there’s a tense moment where Boromir muses its size and importance, and later Saruman calls a storm and the travellers are stranded and nearly killed in an avalanche.
Gandalf asks Frodo to decide their course, and against his advice the Fellowship will travel through the Mines of Moria.

How did it happen in the book?
The Fellowship travels the lands outside Imladris for two weeks in cold wind and constant dusk, before they reach the borders of Hollin, where the weather changes in their favour. There they rest a while and Gimli tells his companions about the mountain range which has came into view while travelling: he names the three most important peaks of the Misty Mountains in Elvish and Khuzdul (the secret language of the Dwarves) and speaks about Khazad-dûm and Kheled-zâram. Gandalf replies by announcing that they will try to cross the Mountains by taking the Redhorn Pass – Redhorn is the Westron name of Caradhras.
Aragorn is worried about the silence that lies in the country, and later that day a flock of black birds appear on the horizon during Sam’s watch, which the Ranger identifies as crebain from Dunland and Fangorn. The Fellowship has to seek cover for the rest of the day. When they set out again at dusk, something flies by unseen but not unnoticed.
The company then decides to attempt the Pass of Caradhras – against the odds that bad weather and spies will prevent them from crossing.
They come across a snowstorm, which nearly costs them their lives, were it not for the wood Boromir suggested to gather and the miruvor they got as a gift from Elrond.
When the storm dies down, Boromir and Aragorn create a narrow path through the man-high snow and carry the Hobbits down.

At the beginning of Chapter IV, A Journey in the Dark, the Fellowship have to discuss their next step. Frodo doesn’t want to return to Rivendell, Saruman’s spies watch the Gap of Rohan and the Pass is closed to them. Gandalf’s proposal is to take the route through the Mines of Moria, but Aragorn and Boromir strongly oppose to this.
The decision is made for them when they hear the howling of wolves, and later are attacked by a horde; but Gandalf uses a few of his lightning tricks to scare the beasts off.

As you can see, a lot was left out in the movie, and those parts of the book which were used, were altered at different points. Let’s take a closer look at those scenes:

The Rest at Hollin
The Fellowship doesn’t journey by night, and so no one is sleeping when the crebain fly over the camp. As a matter of fact, the film makers use this moment to give us a glimpse of what life is like for the travellers: we see Gandalf and Aragorn smoking a pipe, Sam is cooking and Boromir is trying to teach Pippin and Merry some sword fighting.
We also hear Gimli suggesting to Gandalf that they’d take the route through Moria in stead of attempting the Pass. This is very different from the books, where we see Gandalf suggest going through the Mines and Aragorn preferring Caradhras. Gimli is eager to go through the Mines but doesn’t expect his kin to be alive; the long absence of news from Balin is one of the reasons why the Dwarves travelled to Rivendell.
The Fellowship spends the rest of the day in hiding, in the books, and wait for the cover of night to journey on.

Braving Caradhras
The scene with Boromir and the Ring is nowhere in the book: there is no instant where Frodo loses the Ring or drops it somewhere for one of his companions to pick it up; and Boromir only sees the Ring once, at the council, and only from afar.
In the movie, Saruman uses spells to rouse the Mountain and thus causes the avalanche. His aim is to prevent the Fellowship from passing safely, and sending them through the Gap of Rohan where his spies and allies will be able to take the Ring. His guess is that Gandalf will not go through Moria, because he knows that the Balrog lives there.
In the books, however, the Fellowship doesn’t know for certain what is causing the snowstorm they’re caught in: Boromir insists that Sauron is behind it (and Gandalf seems to agree on this), whilst Gimli and Aragorn suspect the mountain itself is simply evil.

Choosing the New Route
As explained above, the positions in this whole debate have been altered: originally it’s Gandalf that insists on going through Moria, and Aragorn who is very reluctant and presses the wizard to attempt the Pass of Caradhras first.
When it becomes clear that they can’t pass the mountain in the movie, Gandalf leaves it up to Frodo to decide. In the books, the Fellowship discusses this matter after leaving the mountain-slope, and a fourth option is added: next to the Gap of Rohan, returning to Rivendell is also named as an option, though dismissed by Frodo who feels it would be admitting defeat.
The final decision is made when the companions discover wolves surround them, and as Boromir says, ‘the wolf that one hears is worse than the Orc that one fears’.
A similar change occurs at Amon Hen, where originally Frodo seeks solitude to decide the Fellowship’s course.

Mistakes
– As Rivka so eloquently put it: Where the heck does Bill the Pony go during the Crebain from Dunland scene?

– Telquellewen_greenleaf pointed out that Aragorn releases the hilt of his sword twice during the confrontation with Boromir.

– Caradhras is not called the Redhorn for no reason: Tolkien describes it as ‘a tooth tipped with snow; its great, bare, northern precipice was still largely in the shadow, but where the sunlight slanted upon it, it glowed red.’

– Gandalf doesn’t know what the Dwarves awoke in the depths of Khazad-dûm. As a matter of fact, when faced with it outside the Chamber Of Mazarbul, he doesn’t even recognise it as a Balrog. It’s only later that he can identify their foe.

Borrowed Lines
– In the movie it’s Legolas that identifies the birds as crebain – Aragorn’s line in the book is ‘They are not natives here; they are crebain out of Dunland and Fangorn’.
The conversation that precedes this (Gimli answering Sam’s question with ‘Nothing, it’s just a wisp of cloud.’ and Boromir replying with ‘It’s moving fast, against the wind.’) is derived from a conversation later that day between Aragorn and Gandalf, when ‘something’ passes them high up in the sky.
Gandalf says that ‘it may be nothing, just a wisp of thin cloud’, to which Aragorn replies: ‘It was moving fast then, and not with the wind.’

– The scene where Boromir picks up the Ring is not in the book; the lines he speaks, however, are. In ‘The Breaking Of the Fellowship’ [2.X.] Boromir says to Frodo: ‘Is it not a strange fate that we should suffer so much fear and doubt for so small a thing? So small a thing!’
A little later, when Frodo tells him he can’t even look at the Ring again, he replies with: ‘I care not.’

– Legolas’ ‘There’s a fell voice on the air.’ is borrowed from Boromir’s comment about the icy wind that comes down Caradhras and hinders their progress: ‘Let those call it the wind who will; there are fell voices on the air; and these stones are aimed at us.’
Boromir believes that Sauron is behind the snowstorm and rockslide.

– Saruman’s comment about the Dwarves of Khazad-dûm, is in fact a line from Gandalf, which he uses in ‘A Journey In the Dark’ when he tells about the history of the Mines: ‘They delved too greedily and too deep.’ [2.IV]

– Boromir’s ‘This will be the death of the halflings’ is indeed a line he uses during the snowstorm, when he has to lift Frodo out of the snow piling up around him.

Bookie Details
– Sam is the first one to spot the crebain, both in the movie and the book.

– Although some complained that it was hardly noticeable, Legolas does walk on top of the snow while the others struggle to get on. He’s also the only one not suffering from cold. This hardly makes up for the loss of ‘I go to find the sun!’ though.

– Boromir and Aragorn both carry two Hobbits up the hill, whilst in the books they each carry two Hobbits down the hill after the snowstorm.