The old ones were more about the science fiction itself, where as the new ones are much more action oriented. They come off as a lot cooler also.
I actually think the new Kirk is just as good as the old kirk. He's convincing as a hot-headed, yet very intelligent and sensible at times, type. They do a good job of showing Kirk grow up in this new series.
The new Spock, isn't quite as good as Nimoy though. Nimoy just seemed much more of a bad ass. A sort of effortlessness about him too. Though the new one has more of a temper problem, he also seems much more arrogant, in a sort of trying too hard type of way.
The new scotty, is just much funnier than the old one. The new Uhura is hotter than the original, and much more relevant to the story IMO. She's boning Spock, which I don't think ever happened in the original.
Bones is probably the most different. The new one is played by a younger, hipper dude, than the old one ever was.
For the most part, the old Star Trek movies were much more 'Trekkie' like. Somewhat dorky, kind of cool, with a little more science fiction, and seriousness behind them.
The new ones, though also very smart, are catered to this generation, and play up the hipper themes in movies. A little more humor, with a bit of the ultra serious at times too. The special effects are WAY better obviously.
Personally I like them both.