Not quite the same thing. According to the premise, these are supposed to be all the same people, just in a different timeline which was changed 25-30 years prior. It's not supposed to be a totally separate continuity. And Khan, being from the 20th century, would not have been affected by the timeline change at all. He's a Sikh.
I really didn't like the first one and thought the premise was really crappy to begin with so honestly I don't expect much of any "continuity". It's just a crappy "we don't have a good explanation, so we did time travel!"
IMO, the time travel added nothing to the movie at all. They should have just skipped the time travel and rebooted, clean. New cast, new story, deal with it and have fun no stupid attempt to shoe horn it all in, especially with something as lame as time travel.
I mean, Spock is making phone calls to himself on the regular. Which means he wouldn't be the same spock when he got older, so he wouldn't go back in time to make phone calls to himself...
I"m glad they just ditched the continuity and made a good villain.
Maybe it IS a totally seperate continuity now. That's how time travel works. It's baloney and you can do anything you want.