From the point of view of formal logic, having a multiplication operation makes a rather large difference. If we use first-order logic, and we omit general multiplication, we have the theory of Presburger arithmetic. This theory is known to be logically complete. But if we use Peano arithmetic... or even if we omit the induction axiom and use Robinson arithmetic, then we are working with an incomplete theory. In other words, if we stick only to addition, every (first-order) proposition we can state about the natural numbers can either be proven or disproven. But if we allow multiplication, then there exist statements that can neither be proven or disproven. (And furthermore, it remains incomplete, even if we adopt finitely more axiom schema) I don't know much about what happens when you allow higher-order logic.