The difference for me is that Nadal was phenomenal on clay, but not on the other surfaces at that age. His whole game was tailored for Roland Garros, and without question he was a phenom on arrival there.
What impresses me most about Alcaraz is not only how rounded his game is from a technical standpoint (Nadal's serve was only good for clay back then and he had to do a lot of work on his top spin forehand to make it more effective on grass/hard courts), but it's how quickly he has worked out a winning game on grass. I think he's tactically superior to Nadal at the same age, and by some distance too.
But, I suppose there's an element of Alcaraz standing on the shoulders of giants in it all because you can see so much of Nadal's game in his, along with Djokovic and Federer. I guess that's what the greats do, and the effect they have on the next generation. Everything moves along and gets better when the truly exceptional try new things and set new standards.