It is the Al+3 ion which does the hardening. It really doesn't matter what salt supplies it. It can come from aluminum sulfate or from sodium aluminum sulfate. I suspect that the bottom line determines which one is used. Sodium aluminum sulfate is one of a group of compounds known as alums which are double salts. For example sodium sulfate plus aluminum sulfate. They exist as definite compounds in the solid state but dissociate in water into the component ions.