The advantage of an Axe is that you can hold a sheild while maximizing full power of the weapon. I have Basch weilding an axe and a sheild and a gauntlet for an accessory. Foes with high killing power more than usually are blocked thus my player will not go down so easily as a plain sword weilding one.

I found this very useful when I first fought Pylraster. His 2000-4000hp hits were being blocked a lot by Brasch while Vaan kept dying.

That's why when fighting high HP foes I always have Brasch in their with Vaan who is the swordsman.