I think they're excellent modernized, moviefied versions of the TOS characters. The biggest difference is that they see each other as friends first and officers second.
Kirk is based more on the classic movie Kirk than the (to be honest, rather generic and occasionally unpleasently grabby) TV series version.
Spock has probably changed the most of the main cast, although after what he's lived through it's not surprising. He's snarky and a little unstable.
McCoy is McCoy, but they're feeding him better in this timeline.
There's a lot more to Uhura. Like the "Mr. Adventure" scene from STIII expanded into a whole person. No more "I'm frightened, captain" and screaming. And she can
Scotty has a bit of a chip on his shoulder now, and due perhaps to the Temporal Cold War has less hair than his older self.
Sulu's speech in ID was the coolest thing the character ever did. Less goofy than TOS Sulu, although we haven't seen enough of him to really say.
Chekov is the same as ever, but a little more humble and perhaps a little smarter - but like Sulu, we haven't seen enough to be certain.
Khan and Carol seem the most different to me, but the both were seen in a totally different situations than before. TOS Khan would probably do the same as ID Khan if put into the same situation. Carol seemed wide-eyed and innocent in ID but world weary in WoK. Such things come with age, I guess...