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.