You get Grand Company seals from them, which can be used to purchase equipment. Further, the most powerful FATE mobs (i.e. Behemoth and Odin) reward special tokens that can be used to purchase equipment.
Aside from that, I think equipment drops are a bad idea. FATEs have already become the premier method of leveling up - to the exclusion of dungeon instances and battle leves. Forcing players to participate in activities other than FATEs from time to time helps them to learn the skills that they need to be useful in the post-storyline content. I'm already hearing horror stories from my FC members about pick-up Trials and Dungeon Instances with players who had no clue what they were doing with their character.
