One slightly strange way you could think about the difference between vultures and axemen, is that a vulture is like an axeman that has been given an inherent combat 1 and 2 but had a Shock promotion subtracted from it.
On defense against melee units, a vulture is equal to an axeman when the total defense modifier is 50%, ignoring the inherent melee bonii. Note this defense modifier includes all attack modifiers except for combat promos. Above 50% the vulture does better, below 50% the axeman does better. So out in the field, vulture will often be weaker defending against axemen than regular axemen because it would be difficult for a vulture to gain 50% of defense other than its melee bonus. But a forest would be enough to ensure it's at least as strong as an axeman defending. Defending a forest, both an axeman and vulture would be 7.5 against a melee unit. Against anything other than melee, the vulture fares better for obvious reasons.
On the attack it's a bit more complicated, but not by much. Obviously the vulture will be better when attacking anything other than melee units, and this is something to strongly consider.
When attacking a melee unit however, an axeman would be better when the sum of defense modifiers is less than 25%, and the vulture better if the defense modifiers are greater than 25%. So already we can say that if you are attacking a hill city, a vulture is no worse attacking that city than an axeman.
So actually it's pretty rare for vultures to perform worse than axes. I think people are misunderstanding things a bit if they think Gilgamesh would get ripped to pieces by enemy axes. The vulture is no dog soldier. If you are being attacked by axes, putting your vultures on good defensive terrain (like a forest) instantly removes its disadvantage of having the weaker melee boost.
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