I think, Echo aside (which is a separate but very known thing that we don't need to discuss), the main difference is aetheric capacity. Ancient souls (and presumably bigger boys like dragons or aliens like Gilgamesh) have a lot more aether to work with; however, that doesn't mean they're inherently superhuman, because they still need to learn a way to use that aether. The devs mentioned in the latest live letter that someone born with an Ancient soul wouldn't be necessarily all that different, but might be a bit more powerful. I take that to mean that most techniques you can learn in the Source just aren't designed to make use of more aether than your average Joe can use, so if you teach someone with an Ancient soul thaumaturgy you're not gonna see someone all too much stronger.

The main soul-related question bouncing around my mind lately is the realization that, if souls are cycled within the lifestream of the nearest planet (which is usually the one they died on), somewhere out there are regular-ass people with Bahamut and Ratatoskr's souls.