Full Elixirs generate 100 AP on use. You do not lose any excess AP that would go over your maximum AP; 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 100 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.