The purpose of mana is to the following:
To allow you to cast a lot of spells in a short window of time if you want to, but to be restricted in the long run. This is especially true in lane. If you want, you can increase the size of the pool, to extend the time till you run out, or chose to boost your regeneration speed.
Energy, on the other hand, is there to restrict you in the short run, but not restrict you in the long run. Meaning, an energy user is worse of in an extended fight, but better in short, hit and run trades.
Health costs serve to make your casting tied to your primary resource, punishing improper usage of abilities more. Usually found on champions who have a way to restore HP to combat their costs (
) or other ways to protect their primary resource (Always shield, never die. Hue Hue Hue Hue.).
Champs who are completely without resources, allowing a playstyle of casting without thinking about secondary costs (primary cost is cost of opportunity, aka, shit, NOW I'd really need that dash to get away/damage to get that last hit) need to be balanced otherwise. For example, Riven has to use her shield sensibly to prevent damage, as her low base regeneration makes all damage really stick. Katarina has no CC and low base damages and scaling if she can't get a reset. Garen has no sticking CC, long CD ultimate and is pretty kiteable.