How did it happen in the movie?
The Fellowship leaves Lothlórien in boats given to them by the Elves. They have to travel the river Anduin to get to Mordor.
During a rest one night, Boromir spots a log floating along the river, which is steered by a strange creature. Aragorn tells him that this is Gollum, and that it has been following them since Moria.
The two Men then start arguing, and Boromir accuses Aragorn of being prejudiced against Men, the inability to see the good and the strength in them.
Sam tries to comfort Frodo, but it is clear that Frodo is already thinking about leaving the Fellowship and going off on his own, as Galadriel advised him.
When the Fellowship takes to the river again, Legolas spots something following them along the banks.
Before the company pulls to shore for a short rest at Parth Galen, they have to pass the Argonath, the statutes of the great Kings of Gondor.

How did it happen in the book?
Tension rises when the Fellowship starts to journey by boat across the Anduin: Boromir is starting to show some strange behaviour, and Sam catches sight of a log with eyes following their company at a certain distance. He tells his master, and Frodo admits that he also saw this creature before, and that he suspects it to be Gollum.
Later that night, during Frodo’s watch, the creature sneaks up on him and Frodo has to draw his sword. Aragorn confirms that Gollum has been following them since Moria.
When they are nearing the Rapids of Sarn Gebir, they are attacked by Orcs but manage to escape thanks to the darkness of the night and the elvish cloaks they are wearing. Only minutes later, when Legolas has reached the riverbank, a great birdlike creature appears in the night sky above them. The Elf manages to scare it off by shooting an arrow at it with the bow he was given by Galadriel, but the company is left with a feeling of dread.
The next day Aragorn decides that they will make for the High Seat of Amon Hen by using an old portage-road, and so travel on to the falls of Rauros. Later in their journey they will pass the Argonath, and Aragorn will undergo an apparent change in the face of his great ancestors, the great Kings of Gondor: Anárion and Isildur, the sons of Elendil. The journey by boat will have taken up ten days.

Let’s take a closer look at the changes in this sequence:
– It’s not Boromir that spots the log-with-eyes, but Sam, who informs his master of his discovery. Aragorn gets involved ‘by accident’, because Frodo’s confrontation with Gollum during his watch awakes him.
– There is no moment in the book where Aragorn and Boromir get so personal: although Boromir often speaks of the valour of his people, insisting on travelling to Minas Tirith, and Aragorn always dismisses of this idea; they never discuss this in private. As a matter of fact, of the four instances where the two Men have a more or less private conversation in the movie (in Rivendell, in Lórien, here and when Boromir dies), only the last one occurs in the book.
– There is also no such moment between Frodo and Sam, where Frodo hints at what he is planning to do.
– Aragorn doesn’t shift into King-mode when he sees the Argonath, unlike in the books where Tolkien describes how a sudden change came over him and he looked as powerful as those Kings of old.

Mistakes
– In the panoramic shots we see during this sequence, there are always trees (a rather large forest, as a matter of fact) surrounding Anduin. However, the Fellowship passes the plains of Rohan, with vast open spaces on either side of the River.

Borrowed Lines

Bookie Details
– The arrangement of the boats is exactly as described in ‘Farewell to Lórien’: Aragorn with Frodo and Sam, Boromir with Merry and Pippin and Legolas with Gimli.
– Aragorn already noticed Gollum in Moria, both in the book and the movie, a sign of his Ranger skills.
– Boromir keeps glancing at Frodo in Aragorn’s boat, as he does in the book.
– Legolas is the first one to spot the Orcs.
– The set designers read Tolkien’s description of the Argonath carefully: ‘The left hand of each was raised palm outwards in gesture of warning; in each right hands there was an axe; upon each head there was a crumbling helm and crown.’ [2.IX]