"I am going to get raped to death by a schizophrenic hobo in a fucking cave. This is not how I saw things ending."
Bilbo appeals to the kinder personality, begging to play the riddle game, "just you and me." If he wins, Gollum will show him the way out. If he loses (both of Gollum's personalities agree), Gollum will eat him. Bilbo accepts these terms, sheathes his sword, and dives straight into the riddles. They offer each other riddles, each getting progressively more difficult. Gollum continues to grow angry, at one point forcing Bilbo to draw his sword in self defense. He eventually picks up a rock and menacingly says that Bilbo gets one more question. Bilbo is unable to think of anything, so Gollum screams, "Ask us!"
Christ, Gollum, calm down. It's polite of you to let him have some last words and all, but you don't HAVE to wait if you wanna just kill him.
The Book: Bilbo never follows Gollum to the lake, and never even knows of the creatures existence at first. Bilbo has a seat at the lake's edge, at the end of his wits, while Gollum watches him from a distance. He uses his tiny boat to swim across the lake, and surprises Bilbo, but doesn't leave his raft to do so. In fact, he remains in it for most of the game's duration. He asks Bilbo what he is, to which the hobbit responds, "I am Mr Bilbo Baggins." Gollum, wary of Bilbo's sword, offers a game of riddles in order to get a measure of Bilbo's character. Bilbo agrees, and they go back and forth with their riddles. Gollum eventually starts to get angry and tired of the game, so he gets out of his boat, sits down next to Bilbo Baggins, and says that Bilbo has just one more question.
What difference does it make? Bilbo's sword is not a necessary source of light in the movie. In the book, it's his only way to see anything (except for Gollum's eyes); without it, he'd be blind in the dark. Therefore, he keeps it unsheathed at all times. There seems to be a mysterious source of light in the movie that lets Bilbo and Gollum see one another without the sword's glow. This doesn't really make a huge difference in how events play out, however.
A much more noticeable difference is one I alluded to in my post about Gollum; specifically, his split personalities. They not only bicker with one another in the movie in a way that is obviously not in the book, but it's an important part of the movie. The reason Bilbo agrees to the game of riddles is an attempt to pacify Gollum, to placate the "good" side in the hopes that he'll be offered a way out, instead of being attacked by the "bad" side.
...or whatever is up with this side.
My Opinion: The inexplicable source of light in the cave is kind of annoying, but it's the sort of thing that plays out often enough in movies that it doesn't matter much; it's not like it's necessary for Bilbo to only be able to see Gollum by the light of his sword (and, in fact, when he does sheathe his sword in the book, there is no explanation for how he is able to see then, either!). It was worth it to see the light flicker as the goblin died; that was a neat touch that I was happy to see. The split personalities weren't a problem, either; they're part of the overall Lord of the Rings canon, and added a nice bit of depth to both Gollum and Bilbo. I'm always annoyed that the movie cuts out various scenes or lines of dialogue that portrayed Bilbo as a good person; I can't have too much of a problem with a scene that serves the same purpose!
It's important to stress, by the way, exactly how pedantic I am being right now. This is not only one of the best scenes in the movie, but it's also the most faithful as well. It's obvious that Peter Jackson went to a great effort to keep this as close to the source material as possible, and it really shows; you can almost follow their dialogue, word for word, in the book.