Full Elixirs generate 100 AP or your max AP on use - whichever is greater. You do not lose any excess AP that would go over your maximum AP except in the case your AP would go over 999; instead, you will temporarily have more AP than your maximum. AP will resume regenerating as normal once your AP drops below your maximum.
In addition to restoring AP, Full Elixirs can be used during battle to fully restore your party to full health with 100% charge bars. They are also used to rejoin a battle if your party is defeated (with full health and charge bars). However, elixir use is sometimes restricted on specific quests or raids.