Therefore, spear skill only increases damage and armor class, not attack. Axe skill only increases damage, not attack and recovery time. Sword skill (for the sword in off-hand) increases attack and decreases recovery time.

So, when dual-wielding, invest in both skills equally to get the best DPS.

Sidenote 1: it's different in MM7 and 8, where spear skill only effects armor class while sword skill effects damage, recovery time, and attack so clearly it's wrong to invest in spear skill in those games.

Sidenote 2: Spear is better for dual-wielding because the advantage of Axe skill (recovery time) is pointless since you need to increase Sword skill for it instead. If you are not dual-wielding, then Axe eventually beats Spear at high enough skill. It has to overcome the weapon itself dealing less damage, being slower, and losing the +1d* bonus you get from using spears two-handed.