Different actors, purpose of the film, etc are interesting observations from the production POV, but are not in universe. The clear, unmistakable visual evidence is that Kirk, Scotty, Sulu, etc look very physically different than they do in the so-called "Prime" universe. The fact that the elder Spock immediately recognizes them indicates that this Spock comes from a universe where that is what they look like and have always looked like.
We don't know what universe this elder Spock comes from. Nothing in-universe definitively establishes it. But apparently it's not the "prime" universe.
We don't know what universe this elder Spock comes from. Nothing in-universe definitively establishes it. But apparently it's not the "prime" universe.