The way I see it, the ones who have changed the least basically boils down to McCoy, Sulu, and possibly also Captain Pike. As follows:
Wasn't born on Earth, didn't even get to meet his father, elevated to the rank of Captain without all that messy business rising through the ranks on ships like the USS Farragut
(though, yes, I suppose there's nothing to say he didn't serve on it as part of his exams or something, before the later events of the movie took hold). By contrast, the series bible for TOS makes a point of mentioning that Kirk's dad got to see his son become Captain of the Enterprise before he died. I have no doubt that Kirk Prime is a much more secure personality for having had his father's guidance all those years. Kirk Prime was also described as being a exquisite student at the academy, whereas NuKirk seems to be a bit brasher and less likely to be "a pile of books with legs" (as Gary Mitchell desribed Kirk Prime). VERDICT: the same Kirk, but with broader personality issues.
to our Spock Prime, but I'm not sure Spock Prime would have had that kind of relationship with Uhura (they flirted in TOS, but it wasn't serious - I wonder what it was that pushed New Spock and New Uhura into taking it to the next level?). The death of his mother undoubtedly makes an impact on his life though. VERDICT: as with Kirk, broadly the same but with a few 'issues'.
Same as above, really. She's arguably more assertive than TOS Uhura, although "The Search For Spock" shows Uhura Prime in a light that is considerably sassier than anything we got to see her do in TOS itself. VERDICT: Hard to call on this one. She might not have a different personality at all, it's just expressed
better in the Abrams movies.
The same old Leonard 'Bones' McCoy, albeit just a smidge younger than we ever saw him before. Stuff that is only canonised in the new movies (all the stuff about his divorce) was always there as character background in the TOS series bible, it was just never mentioned on-screen (although it was heavily implied at times, and got very close to being canonised in early drafts of "The Way To Eden"). VERDICT: Identical in virtually every way.
Older (by, what is it, about 4 years or something?) than TOS Chekov, so by extension he hasn't got the same birthday. He's also a bit of a whizzkid, which was kind of implied with Chekov Prime but also kind of denied (Chekov Prime was a bit of a knockabout character at times, happier canoodling with pretty girls than doing his job). VERDICT: I'll call him different. He shares an accent with his Prime counterpart, but not really the same personality as far as I see it.
Contentious, but Scotty Prime was a very specific character pretty much created by (and influenced by) Jimmy Doohan himself. Likewise, the new Scotty is more or less just a typical Simon Pegg character, but with a Scots accent. He's obviously got the brushstrokes of Scotty Prime, but I can't really see much comparison between the two. VERDICT: different.
What do we learn about Sulu in the 2009 movie? He's the helmsman. He's oriential. He likes playing with swords. So far, so recognisably Sulu. I'm not sure there's anything
different about him other than he's obviously being played by a different actor. Unless the sequel does something major with his character I don't see any divergence at all from his Prime counterpart. VERDICT: same old Sulu.
he's sketched out a bit more than Jeff Hunter's version, but on the face of it I think he's pretty accurate to how I imagine Hunter's Pike would have been most of the time. We saw Pike in "The Cage" at a particular point in his life, dealing with insecurities and wondering if he made the right career choice after all. But underneath it all, I think they are basically the same character: a noble and heroic architype, who is given the added benefit in Abrams' Trek of also being a surrogate father figure to Kirk. VERDICT: not too different.
Your thoughts, ladies and gents?