Well, the full moon puts out more light. That means you get more ambient light in the water on a night dive IF it's not cloudy/overcast.
The night time predators rely largely on surprise and darkness to catch their prey, so with more light, the potential prey is more likely to see the predators and avoid them. This means the prey will be much more comfortable being out in the open, and you will probably spot more small fish and invertebrate life. It also means you can do your dive with a less powerful dive light. The light tends to scare fish off, since it blinds them and makes them easier prey to predators, so a lot of critters will flee from your dive light. If it's bright enough, it's possible to do a whole dive with your lights off, and you will be able to get a better look at some of the cool stuff.
If you're interested in seeing some of the bigger predators, like grouper and tarpon, they are usually more active when the moon isn't out, and they also are attracted to your dive light since it tends to blind their prey, making them an easier catch.
You also need to account for when the moon rises. There may be a full moon, but it might not rise until sometime after sunset.
I would say the best place for a night dive is anywhere there is a lot of fish life. Also, the most active time on the reefs is right around sunset. You see a lot of the daytime critters going to their hiding spots, and the nighttime critters coming out. Kind of like the time in between shifts in a factory.
Tom