Minecraft wiki entry for Moon, associated text: The number of slimes that spawn in swamps is proportional to the moon's fullness: They are most numerous during the full moon and none spawn during the new moon. (The moon's phase does not influence the normal spawning of slimes in designated chunks at altitudes below y=40.)
The fuller the moon, the higher the chance skeletons and zombies have the ability to pick up items and spawn with weapons and armor. Furthermore, there is a higher chance for their equipment to be enchanted, and if it is, the enchantment levels are higher. A fuller moon also gives spiders a greater chance to spawn with random beneficial status effects, including invisibility‌[Java Edition only]. All of these effects combine with the world's difficulty, and higher difficulty levels increase the chance of mob buffs.
During a full moon, 50% of cats spawn as black.