Both expressions say the same thing.
But note that saying "$f: A \to B$ is a function from $A$ into $B$ does not imply that $f$ is into but not onto (i.e., it does not rule out that $f$ might be onto, so it is not the "converse" of, or the negation of, the descriptor "onto" or "surjective"). Indeed, it says nothing more, and nothing less, than the alternative: "$f$ is a function from $A$ to $B$."
So we can either say that there is no such thing as an "into function" (i.e. it is NOT used to describe some, but not all functions, nor is it describe particular kinds of functions),
Or we can say that every function is into (meaning every function $f: A \to B$ is a function on $A$ into $B$.)