From the dictionary definition of a fable:
A short story, typically with animals as characters, conveying a moral
Oxford Dictionaries: "Fable"
Merriam Webster defines Fairy tale as:
a story (as for children) involving fantastic forces and beings (as fairies, wizards, and goblins)
a story in which improbable events lead to a happy ending
Merriam-Webster: "Fairy tale"
The defining factor of a fairy tale seems to be a story involving fantastical beasts and their magical world.
A more opinionated difference: There is no hidden meaning in a fable, and it is meant to convey a moral truth. On the other hand, fairy tales weave a world of magic and leave much room for imagination and fantasy. It is up to the reader to decide if he grasps any message or just enjoys the imaginary world of fairies and elves.
Taken from differencebetween.com: "fairy tales and fables"