The most notable differences are in the Riddles in the Dark chapter. In the original, Gollum willingly offered up his ring as the prize in the riddle game, which isn't compatible with his later life-altering obsession with it.
Can you imagine this making any sense now?
I don't know how many times Gollum begged Bilbo's pardon. He kept on saying: "We are ssorry; we didn't mean to cheat, we meant to give it our only only pressent, if it won the competition." He even offered to catch Bilbo some nice juicy fish to eat as a consolation.
Some of the lines that aren't changed undergo substantial reinterpretation.
"Where iss it? Where iss it?" Bilbo heard him squeaking. "Lost, lost, my preciouss, lost, lost! Bless us and splash us! We haven't the present we promised, and we haven't even got it for ourselves."
"Where iss it? Where iss it?" Bilbo heard him crying. "Losst it is, my precious, lost lost! Curse us and crush us, my precious is lost!"
Consider "Lost, lost, my preciouss, lost, lost!", which is the same in both versions. But in the first one, it simply refers to a lost object, aggravating but not all-consuming. In the second, it's not just the ring that's lost, but all of himself. It's not frustration, but utter despair.
In the new version, Gollum is more evil, but also more pitiable. It's worth reading both, next to each other to see the differences:
I'm also gonna put in a plug for the book by Corey Olsen, who spends a lot of time on this question. His analysis is fascinating and compelling, and I recommend it to everybody who wants to take another look at The Hobbit:
Bonus factoid: in the introduction to Fellowship of the Ring, the first edition of The Hobbit is retconned into a lie by Bilbo, trying to justify his own theft of the ring, while the revised version is more "factual." And the "lie" is a symptom of Bilbo being overcome by the Ring! This is the kind of thing Tolkien did all the time, setting himself a puzzle and then solving it.