The scene in the film comprises only a small amount of the chapter “The Riders of Rohan” in the book, which starts with the Three Hunters finding out the Orcs are taking the hobbits to Isengard, and ends with them seeing a shadowy figure at the edge of Fangorn Forest.

Meeting the Riders

The scene in the film starts with one of Legolas’s pronouncements, “The red sun rises, blood has been spilled this night”, clearly paraphrased from a section earlier on in the book chapter: “As before Legolas was first afoot, if indeed he had ever slept. “Awake! Awake!” he cried. “It is a red dawn. Strange things await us by the eaves of the forest. Good or evil, I do not know; but we are called. Awake!” [3.II.]
Then the riders appear in both film and book (though a day and many miles go past between Legolas’s mention of the red dawn and the meeting in the book).

In the film, Aragorn hears horses and gestures to the others to hide with him behind a rock. But when he sees who the riders are, he stands and yells out to them, asking for news from the Mark.
Needless to say, in the book, the three hunters do not hide behind a rock, and the riders do not come upon them so suddenly. Aragorn first sees them as a shadow on the distant green, five leagues distant, and Legolas then looks and sees 105 horsemen (“and the glint of morning on the tips of their spears was like the twinkle of minute stars beyond the edge of mortal sight” [2.II]). The three of them then walk a little further and sit and wait for the horsemen to come near, hoping for news of Merry and Pippin.

While waiting, Aragorn tells the others of the history of the Rohirrim, calling them proud and wilful, true-hearted, generous in thought and deed, bold, wise but unlearned. Gimli mentions that Gandalf spoke of a rumour that the Horse-lords may pay tribute to Mordor.

The riders then approach, with some of the most wonderful descriptions of the Rohirrim in any of the books: “a long line of mail-clad men, swift, shining, fell and fair to look upon.
Their horses were of great stature, strong and clean-limbed; their grey coats glistened, their long tails flowed in the wind, their manes were braided on their proud necks. The Men that rode them matched them well: tall and long-limbed; their hair, flaxen-pale, flowed under their light helms, and streamed in long braids behind them; their faces were stern and keen. In their hands were tall spears of ash, painted shields were slung at their backs, long swords were at their belts, their burnished skirts of mail hung down upon their knees.” [2.II.]

So how closely does the film manage to match this detailed description? Well, it manages *quite* well. Not all the horses are greys (though most are), and they do not have their manes braided. Some of the Rohirrim aren’t exactly flaxen-pale, including Éomer. The spears don’t really look like ash and the pennants of the film aren’t mentioned in the book, but all in all, the Rohirrim are pretty similar.

From there, the encounter flows similarly. Aragorn calls out to the Horse-lords in each case, and the Rohirrim encircle the three hunters, spears pointing at their faces. Éomer then rides forward and asks who they are. As with the Rohirrim in general, film Éomer fits the description in the book rather well (“a tall man, taller than all the rest; from his helm as a crest a white horsetail flowed”).

In the book, Aragorn then readily gives his name as Strider on Éomer’s demand, while in the film, Gimli causes a diplomatic incident immediately refuses to name himself until Éomer does so as well. Aragorn has to calm his companions and gives his name – this time as Aragorn son of Arathorn.
Éomer, in the book, then talks to Aragorn, saying that he had initially thought them Orcs, but now realises his error – promptly asking if they were elvish folk. Aragorn then introduces Legolas and Gimli, mentioning that they had been travelling through Lothlórien, and that they carry the gifts and favour of its Lady.
Here something is brought out in the book that isn’t made so obvious in the film – the way Men consider Lothlórien as a dark and treacherous place, ruled by a elven sorceress: “The Rider looked at them with renewed wonder, but his eyes hardened. “Then there is a Lady in the Golden Wood, as old tales tell!” he said. “Few escape her nets, they say. These are strange days! But if you have her favour, then you also are net-weavers and sorcerers, maybe.”” [2.II]

He then asks Gimli and Legolas why they are so quiet, and Gimli replies with a line taken almost word for word into the film: “Give me your name, horse-master, and I will give you mine, and more besides”. [2.II.]
Book Gimli then proceeds to cause a similar diplomatic incident, telling Éomer what he thinks of anyone believing in the dark tales of the Golden Wood, and especially anyone who dares malign Galadriel: “”You speak evil of that which is fair beyond the reach of your thought, and only little wit can excuse you.”
Éomer’s eyes blazed, and the Men of Rohan murmured angrily, and closed in, advancing their spears. “I would cut off your head, beard and all, Master Dwarf, if it stood but a little higher from the ground,” said Éomer.
“He stands not alone,” said Legolas, bending his bow and fitting an arrow with hands that moved quicker than sight. “You would die before your stroke fell.”” [2.II]
I am sure all will recognise that set of lines from the film.

Discussion

In the film, after hearing their names, Éomer proceeds to tell them all about Théoden and the bad times in the Riddermark, saying also that the white wizard walks abroad as an old man, hooded and cloaked. In the book, everyone is more cautiously, slowly trading pieces of information as they become more sure of themselves. Aragorn starts the parley by asking Éomer whether they are friend or foe of Sauron (Éomer answers that they do not serve the Black Land, but neither are they at open war with him) and then decloaks himself – literally.

“Aragorn threw back his cloak. The elven-sheath glittered as he grasped it, and the bright blade of Andúril shone like a sudden flame as he swept it out. “Elendil!” he cried. “I am Aragorn son of Arathorn … Will you aid me or thwart me? Choose swiftly!”
Gimli and Legolas looked at their companion in amazement, for they had not seen him in this mood before … in his living face they caught a brief vision of the power and majesty of the kings of stone. For a moment it seemed to the eyes of Legolas that a white flame flickered on the brows of Aragorn like a shining crown.” [2.II]
This is a significant loss from the film – Aragorn strong and sure, showing the power of his line and his personality. The nearest Aragorn gets to this in the film is running around yelling orders and slaying vast quantities of Orcs in the Battle of Helm’s Deep

In both film and book, they then get on to discussing the Orcs and the hobbits, Éomer giving them the same news in both:
Éomer: The Uruks are destroyed, we slaughtered them during the night.
Gimli: But there were two hobbits. Did you see two hobbits with them?
Aragorn: They would be small, only children to your eyes.
Éomer: We left none alive. We piled the carcasses and burned them.

After that, the Rohirrim in the film depart, leaving Hasufel and Arod for the Three Hunters to ride, and with a typically doom-laden parting line: Éomer: Look for your friends, but do not trust to hope. It is forsaken in these lands. We ride north!

In the book, the discussion continues further. While Éomer believes their tales of hobbits (again another myth and fantasy for the Rohirrim), Éothain (another scandalously omitted character) refuses to accept Aragorn’s words and wants to tie the Hunters up and take them to the king.
As a mark of trust, Aragorn then tells Éomer more of their journey, from their leaving Imladris onwards. He tells of Gandalf, and Éomer warns him not to mention the Istar to Théoden, for the king had not forgiven Gandalf for recently “borrowing” Shadowfax. He tells of Gandalf’s fall, and Boromir’s death, and when he tells Éomer how far they have travelled, the Rohir names him Wingfoot.
Éomer then tells Aragorn more of their own situation. That the East-mark is his responsibility, that they will hold their ancient alliance with Gondor, and that he has to return to Théoden. That the Rohirrim have no dealings with Sauron, for he treats his horses evilly, and that their chief concern was Saruman. Also that he suspected a league between Orthanc and Barad-dûr. Then he asks that Aragorn will aid them: “”Will you not come? Do I hope in vain that you have been sent to me for a help in doubt in need?”
“I will come when I may,” said Aragorn.
“Come now!” said Éomer. “The Heir of Elendil would be a strength indeed to the Sons of Eorl in this evil tide.”” [2.II]

Éomer continues to talk about the fey and the strange, a superstitious side to the Rohirrim which is completely missing in the film: “The world is all grown strange. Elf and Dwarf in company walk in our daily fields; and folk speak with the Lady of the Wood and yet live; and the Sword comes back to war that was broken in the long ages ere the fathers of our fathers rode into the Mark! How shall a man judge what to do in such times?” [2.II]

Eventually, Éomer then lends the Three Hunters Hasufel and Arod, asking only that when their quest (for the hobbits) is achieved or proved vain, to return with the horses to Meduseld – both for the good of Rohan, and his own standing with the king: “Thus you shall prove to him that I have not misjudged. In this I place myself, and maybe my very life, in the keeping of your good faith. Do not fail.” [2.II]
That one sentence sums up a lot of differences between film and book Éomer.

Towards Fangorn

The last part of the scene involves the Hunters riding towards Fangorn. In both film and book, they ride to the pile of dead orcs, and in the film Gimli finds one of the hobbit’s sword belts. Assuming that they both have died, Aragorn and Legolas grieve (Legolas in muted elven way, Aragorn in a more dramatic and loud fashion). In the book, however, they find nothing belonging to Merry and Pippin, and simply set up a camp near the pile of dead Orcs and the barrow raised for the fallen Rohirrim.

In the book, one last event occurs. As the hunters are settling down to sleep: “just on the edge of the firelight stood an old bent man, leaning on a staff, and wrapped in a great cloak; his wide-brimmed hat was pulled down over his eyes
…
“Well, father, what can we do for you?” said Aragorn, leaping to his feet. “Come and be warm, if you are cold!” He strode forward, but the old man was gone.”” [2.II]

They then discover that the horses were gone, having dragged their pickets and disappeared. Legolas thinks that the figure was Saruman: “Remember the words of Éomer: he walks about like an old man hooded and cloaked. Those were the words. He had gone off with our horses, or scared them away …
… [Aragorn] “But I marked also that this old man had a hat not a hood. Still I do not doubt that you guess right” [2.II]