Another explanation for the STiD Khan looking so different that the TOS Khan: they are two different people.
The film never actually stated that this John Harrison character was Noonien Singh. The character only refers to himself as "Khan", never "Noonien" or "Singh". The only character who refers to him as Noonien Singh in the film is Spock Prime, and Spock Prime was really only talking about the Khan that he knew from his timeline, not having any details about this particular Khan. Remember, "Khan" is a title.
Perhaps this particular superman was just on the Botany Bay, but wasn't actually Noonien Singh. He could have been another of Singh's followers who just took the title because Singh was no longer alive or active to hold the title. The problem with this theory, of course, is the numbers. "Space Seed" established that there were 85 supermen on the Botany Bay originally, but 12 cryogenic chambers failed, leaving 73 (including Khan Noonien Singh), with Singh being the first to revive. So in the JJverse alternate timeline it would again make sense for Singh to be the one reanimated first, with the other 72 staying in cryo. And that is apparently the backstory of this movie.
But there are ways around that. For instance, the Botany Bay was found over 5 years earlier in this timeline than in the Prime one. Perhaps one less cryo unit had failed in that time, meaning 74 were still active when the Botany Bay is found. Singh awakes first, but is killed by Section 31 agents for some reason. [Or Singh dies upon reanimation; remember that almost happen in “Space Seed” and only didn’t happen because McCoy and Kirk were there to help.] Or maybe Section 31 put Singh back in cryo because he was hard to control. Then Section 31 reanimates a different superman, maybe one they thought would be easier to control. This new superman learns that Singh is dead and takes on the leadership role of "Khan" for the 72 still stuck in cryo.
Anyway, all this just means that the new movie version of Khan doesn’t have to have any connection or correlation with the Khan we’ve seen in prior Trek or the novelverse.