How did it happen in the movie?
When the Fellowship arrives at Parth Galen, both Boromir and Frodo are seriously at unease. During the rest, Frodo encounters Boromir who is collecting twigs and wood. The Man once again asks him to reconsider, to lend him the Ring; but when Frodo refuses he looses his temper and attacks the Ring-bearer.
Frodo manages to escape by putting the Ring on, and flees to the High Seat of Amon Hen. There he sees the tower of Barad-dûr, and the Eye of Sauron reaches out for him.
After he’s taken the Ring off, he encounters Aragorn who resists the temptation of the Ring, and understands that Frodo has to go on alone.
Frodo then has to run from the invasion of a massive amount of Uruks, and manages this with the help of Merry and Pippin: Merry, who seems to understand what Frodo is planning, creates a diversion for the Uruks.
When Frodo is already halfway to crossing the river, Sam pops up out of nothing and insists on staying with his master and wades in. Because he can’t swim he nearly drowns; but Frodo saves him.
At the end of this sequence, also the end of the movie, we see Frodo and Sam overlooking the Emyn Muil. Frodo thinks about the wellbeing of his companions, and as ever, Sam manages to lift his spirits a little.

How did it happen in the book?
After the Fellowship has taken a rest at Parth Galen, Aragorn urges the company to decide the next step in their journey. He lays the burden on Frodo’s shoulders, who asks a few hours on his own to come to a decision; while the others stay and discuss the matter amongst themselves.
But Boromir follows Frodo and tries to persuade him to lend him the Ring to defend his city and defeat Sauron. When Frodo keeps refusing, Boromir looses his temper and attacks the Hobbit. Frodo manages to escape by using the Ring, and flees to the High Seat. When he takes place there, he gets a mighty vision of armies gathering on all corners. Then he sees the Eye, searching for him, and he only barely manages to take the Ring off.
In the meantime, Boromir has returned to the camp and the others start a search for Frodo. Sam is told to stay with Aragorn, but guessing his master’s mind he goes to the boats and finds an empty boat floating on the water. He follows it, but almost drowns. Frodo, still invisible, saves him and turns the boat so they can return to shore. There Sam takes his pack, and crosses the Anduin again with his master.
When they reach the other side, Frodo expresses his fear for his companions, but Sam manages to speak some comforting words to him.

Let’s take a closer look at the changes in this sequence:
– Frodo seems to just wander off, in stead of leaving the company to make his own decision. This moment, where Frodo has to decide the fate of the others, was moved to the Caradhras-sequence in the movie.
And Boromir wasn’t gathering firewood, but followed Frodo deliberately.
– After Frodo falls off the High Seat, he meets Aragorn. This doesn’t happen in the books: Aragorn, looking for the Ring-bearer, climbs the hill of Amon Hen and takes the High Seat himself, but doesn’t see anything. Frodo is already long gone by that time.
– Frodo doesn’t encounter Merry and Pippin, and it was Pippin that had already guessed what Frodo was planning to do.
– In the books, Boromir returns to the camp and informs the others of what happened (well, only the necessary info, really). He is then assigned to stay with Merry and Pippin, when they set out to look for Frodo.
– Sam doesn’t come across his master accidentally like in the movie, but understands his mind. He doesn’t wade into the water but sort of falls over, and invisible Frodo pulls him up by the hair. After being saved, Sam runs back to get his backpack – this is how Aragorn figures out that Sam was with Frodo when he left.

Mistakes
– Before Frodo leaves Aragorn says they’ll wait till nightfall to cross the lake because there are Orcs there – wouldn’t they be safer in the day, Orcs being afraid of light and all? Both PotbellyHairyfeet and Mara_Jinn asked this just question.

– So, where is Sam’s backpack?

Borrowed Lines
– Sam’s ‘Strider will look after them’ is actually one of Frodo’s lines. [2.X]

Bookie Details
– The ending of the movie is almost exactly the same as the ending of the book: ‘We will go, and may the others find a safer road! Strider will look after them. I don’t suppose we shall see them again.’ To which Sam replies: ‘Yet we may, Mr Frodo. We may.’