My comment for easwaran seems important enough to repeat here for the world.

Axiom system `A` is more useful than axiom system `B` when `A` correctly proves more recursive functions are total. Axiom system `A` is somewhat more useful than axiom system `B` when `A` correctly proves more turing machines do not halt.

Of course this is all on the fringe of usefulness. Most people just care if P (or maybe NP) algorithms are total, and I believe PA is more than sufficient for this task.