It was made today. Last year in fact. It was called Star Trek XI.
Exactly so.
Based on that but mainly on common sense there are a couple of obvious differences:
The age of most of the characters would be younger, with one or two (most likely McCoy or Scotty) older than on the original series.
Uhura would have a more substantial, important part on the series and at least one of the other regulars would be a woman - most likely Spock.
This is very sensible. One of the reason of TOS popularity it's because its characters are very archetypal.
Kirk won't be very different as the action hero: a white young man burdened with responsibilities beyond his age
Spock would be most likely a black or a woman, in the same vein many especially smart characters have been portrayed recently (even if I always thought that Nimoy's Jewishness worked very well for Spock). There will be sexual tension with Kirk (open if heterosexual, as "bromance" if homosexual).
McCoy would be probably older, a cranky old fellow and confindent to hero Kirk. It would have a secret and tragic past.
Scotty would be probably more nerdy/geeky and more of a computer genius than a "miracle worker": I can actually see his as being younger than his classical self. Not sure if he would actually be Scottish: even in that case, I would bet it would be very much downplayed.
Sulu would be actually from Asia and not just San Francisco: I would say he would be more serious and laconic. Another candidate for a tragic past. Another take would be making the character female: in that case I would say they will make her younger and shyer.
Chekov would be more or less as we saw him in the ST09: young, naive and comically Russian. In case Sulu were a girl, the two would be paired.
Uhura would be the one with the most apparent changes, as the role need to be expanded: my money would be that her African heritage would be downplayed or outright ignored, but there will be much emphasis in her ability to kick ass, making her a "tough girl".