Actually, there aren’t apparent differences between these three words. In most situation, you can replace any one word by the another word. If you insist on knowing their differences, I can tell u some example.
people usually say
”别动” when kids want to run around or want to move sth ,which meaning “don’t move!”
“别碰”when kids or adults want to touch something delicate ,expensive or dangerous.("碰"means "touch")
"别摸"is similar with "别碰" ,meaning You can't touch something .the difference from "别碰” is that “摸" means your fingers is brushing something gentlely ,likes touching the skin or face.
Actually, the meanings of these three words is almost same. You don't have to tell them .You can replace everyone by another casually depending on your habits .It's totally ok.
Tips: only “别动” have two meanings , one of which mean keeping still and don't run around.