Well, I'll admit the debate rose out of an uneducated but highly opinionated non-hunter's rudeness.
But one side is that the bow requires you to be closer to the animal, which may improve accuracy, and it slices the target, allowing better penetration and blood loss. The other side is that the gun's projectile has much more speed (and therefore energy) and does more damage, increasing blood loss. This is assuming proper shot placement in either case.