I'd say that the (obvious) difference between Ranged DPS (AD Carry) and Mage/Nuke (AP Carry) is the damage they deal.
AD carries generally build AD, attack speed, and crits for autoattack damage. Some deal pretty bursty damage. Short-term damage steroids like Graves's dash and Tristana's AS boost increase damage output for a short period.
They deal physical damage, and are countered by armor.
There's also melee carries, who are AD but don't fulfill the same role as a ranged carry. They can't assault a tower from safe range, and cannot consistently survive teamfights without completely destroying the opposition/bailing at the right moment. But they have more damage steroids, invincibility periods, or other lulzy stuff that lets them build damage without fear... if only for a while.
AP casters use abilities to deal damage. Most of them build AP, some get CDR. Many have stuns, silences, or other CC - Crowd Control, aka interrupting, spells. Many mages have strong ults as well as good AOE damage.
Almost all deal pure magic damage. Only exceptions are Lichbane (which deals physical damage scaling off AP) or Ahri's Orb, which deals true damage on the way back.
Burst damage is high - ult spells help in bursting down the enemy team before you die. Mages cannot 1v1 targets as easily if they've blown all spells in the fight, but they can stay back while cooldowns refresh.