How did it happen in the movie?
As soon as the Fellowship goes to shore at Parth Galen, Legolas tells Aragorn that he has a sense of unease, but Aragorn doesn’t seem too impressed.
After a while, in which Gimli vividly describes the Emyn Muil to Pippin, the companions discover that both Frodo and Boromir are missing. They set out to find the Ring-bear, and it’s Aragorn who finds him at the foot of the High Seat. He resists the temptation of the Ring, and then single-handedly fights off a rough five dozen Uruk-hai.
Meanwhile, Merry and Pippin have also come across Frodo. Merry understands that Frodo means to travel on alone, and creates a diversion for the Uruk-hai. In doing so, they put themselves in great danger, but things start to look up when Boromir appears with his big sword and blows his horn. The others, who have been fighting a little further up the hill, hear his call and answer it – but too late.
Boromir and the Hobbits fight valiantly, but the Man takes three arrows in the chest and has to watch how the halflings are carried off helpless. Lurtz, the captain of the Uruk-hai, prepares to shoot one last arrow at a kneeling Boromir; but Aragorn swiftly intervenes and manages to kill him.
Boromir, with his dying words, admits that he betrayed the Fellowship and pledges his allegiance to Aragorn, accepting him as the heir to the throne of Gondor. There is much crying and emotional music, and Boromir is placed in one of the boats with his belongings and carried off by the stream.
Aragorn chooses not to follow Frodo and Sam, but in stead to hunt down the Uruks and save Merry and Pippin.

How did it happen in the book?
After a rest at Parth Galen, the Fellowship has to decide what their course will be. Aragorn lays it upon Frodo to decide, and the Ring-bearer seeks some solitude in the woods.
When Boromir, who sneaked out during their talk, returns to the camp and informs them that he and Frodo had some sort of argument, the company splits up in search of the Hobbit. Aragorn tells Boromir to look after Merry and Pippin, and takes Sam along to Amon Hen.
But when he comes to the High Seat, he doesn’t see anything of any significance, but the surrounding lands. He lingers too long, until the horn of Gondor awakes him from his musings.
By the time he arrives at Parth Galen it is already too late: the Uruks that had attacked the camp have fled, taking Merry and Pippin with them and Boromir lies mortally wounded against a tree-trunk. With his dying words, Boromir confirms what Aragorn already suspected: that he had tried to take the Ring.
The three companions that are left, then mourn Boromir’s passing: Legolas and Aragorn sing a lament for their fallen companion; and he is placed in one of the boats with his belongings and the weapons of his slain enemies at his feet.
When they search for Sam and Frodo, they find a boat and Sam’s pack missing. After some thought, Aragorn decides that their only course now is to try and save Merry and Pippin.

Let’s take a closer look at the changes in this sequence:
– Merry and Pippin don’t run away to create a diversion for Frodo to escape. We don’t really know what happens, except that there was a bit of a fight which apparently ended when the Hobbits were taken. Boromir doesn’t come to their rescue, he was with them at the time when they were attacked.
– During the entire fight, Aragorn is on the Hill of Sight, trying to see something spectacular but failing miserably.
– The band of Orcs attacking the Fellowship are not only Uruk-hai – amongst the slain Aragorn finds only four of a breed he can’t identify; the others are ordinary Orcs from Mordor and Moria – the same ones we’ll encounter in The Two Towers.
The leader of these Uruks, the so-called Lurtz, is a character created by the film makers.
– Boromir doesn’t talk so much before he dies, and he certainly doesn’t pledge his allegiance to his King. After his death, Legolas and Aragorn sing a lament and row alongside Boromir’s funeral boat up to the edge of the fall.
– Aragorn deduces from the missing boats and the missing backpack that Frodo went on his own, and Sam accompanied him.

Mistakes
– A lot of people complained about the fact that Lurtz’s chopped off arm doesn’t bleed. This was done, according to the EE commentary, because PJ wanted to keep the rating low.

– In the books, Boromir’s sword is broken at the hilt.

Borrowed Lines
– When the Fellowship arrive at Parth Galen, Legolas says to Aragorn, ‘A shadow and a threat has been growing in my mind’. This is a line Aragorn says to Frodo, and in stead of mind he says ‘sleep’. [2.X]

Bookie Details
– Aragorn’s battle cry is ‘Elendil!’, the name of his ancestor. In this scene, we hear him shout it when he jumps on the Seat of Sight.

– The scene of Boromir’s death resembles the book description very closely: Boromir’s horn is cloven by an Orc arrow, and he was ‘sitting upright as if resting’ with ‘many black arrows piercing his chest’ when Aragorn found him. Aragorn ‘knelt for a while, bent with weeping, still clasping Boromir’s hand. So it was that Legolas and Gimli found him.’

– Legolas searches for arrows amongst the fallen because his quiver is empty.