I think axe and sword is alot like machine gun vs rifle (or sniper rifle).
One is big, powerful, and makes alot of noise, while the other requires more skill, and goes in for a clean, hopefully quick kill.
Although that is of course an exaggeration, both can kill effectively, again its largely situational. Also, at least in most contexts swords have a lower damage level but are far more precise (more likely to hit) because those using swords, theoretically, are better trained, and for those of equal training, the sword would still have more precision.
Of course, no one is arguing against the stopping power of an axe when it actually connects.
I have two new unit lines thought up
Warrior-> Soldier -> Legionnaire -> Champion (Centurion)
Warrior -> Miles -> Soldier -> Champion