It could have been easily explained by having them say Admiral Marcus had his appearance (which should be as well known to everyone as Hitler’s is) changed so he wouldn’t be recognized (as a deleted scene or something explained why his daughter had a British accent). But no, even that simple solution was apparently too difficult for them. The “plot” was just a a couple very thin slices of bread to hold the action scenes together. Amazing that people got paid a lot of money to write that.
I’ve seen some sloppy crap on the soaps (with respect to casting and writing) that I watch, but that would’ve been unforgiveable (and the fans would’ve demanded a change). That was even worse when they changed Ming the Merciless from a vaguely Asian (albeit alien) caricature in the “Flash Gordon” film (and old serial) into just another white guy in a recent series.
Javier Bardem was one of the few people I could think of who could come remotely close to Montalban’s portrayal. I think I told you guys I had a strange dream where the late Ricardo Montalban came to my house for lunch and I immediately started in by asking his opinion of the “Faux Trek” film and recast and he just shook his head, “No. That’s not Khan.”
It's true that a brief throwaway line (that they were too lazy to add into the script once they cast Cumberbatch) would have explained why he looks completely different. However, even the plastic surgery explanation (which works because I'm assuming 23rd century plastic surgery is far more advanced than modern day) doesn't account for the personality difference.
Cumberbatch and Montalbaun were both willing to trample anyone who got in their way in order to preserve their crew, but otherwise they were completely different villains. Cumberbatch was supposed to be "Khan" from the Space Seed TV episode, and that Khan was exotic, suave, charming, and seductive. A key point in the episode was he convinced Enterprise crew members to defect and join his side, and seduced the ship's historian into becoming his lover.
Cumberbatch's "Khan" was just a cold icy psychopath/terrorist hellbent on revenge against Marcus. Basically the writers wanted to cash in on the name "Khan" without being faithful to the character or having a storyline that fit with his background. They even admitted they turned Harrison into "Khan" AFTER they had completed the story, and retroactively tried to make "Khan" fit into a story about a Starfleet officer turned terrorist.
If they were insistent on recycling a classic villain, they could have used one that actually fit such a storyline. If Cumberbatch had revealed his true identity as either Captain Garth of Izar or Dr. Tolian Soran, it would have fit the type of story and actor they went using (white, British accented psychopath who is "better than Kirk at everything" and hellbent on getting revenge against Starfleet in the mid 23rd century)
But they went for "Khan" simply because he was the best known Trek villain, which would sort of like casting Samuel L. Jackson to play an inner city Los Angeles gang leader codenamed "Eddie Smith" in a Bond movie and then revealing his true identity to be "Auric Goldfinger" after Bond captures him.
Epic fail.