If it's an MMO, I don't really care as much as long as I'm putting the numbers out. However, if it's a caster, I'd rather cast fast and have less of a chance to be interrupted. In action games, I also often find a speedy character more effective because of their ability to adapt to a changing situation and get out of a skirmish quick if something bad is about to happen. In Super Smash Bros Brawl, I play Toon Link over Link for the exact reason that Toon Link is so much more mobile.
I'm big on Fire Emblem, and if you aren't one-shotting everything, you always want to have just enough speed to get in two attacks. There are very few instances when two weaker attacks are worth less than one big hit.
As far as shooting games go, stopping power is more important, so you really need some balance of power, accuracy, and speed. There's no real point in favouring a fast-reloading pistol over most rifles and shotguns, though I feel modern games have screwed everything up by making knifes the most powerful common weapons just because of difficulty of execution. But ultimately: Explosions > all.