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Both are categorized as folk literature and both can be passed down from generations but a fable has a moral lesson to be learnt where as a fairy tale is more for solely entertainment

A fable is a narrative whose characters are animals or inanimate objects. They always end with a moral lesson; this means that they teach values to the readers. Fables are not  based on facts.

Fairy tales have persons as characters. These are generally royal members or knights. They are tales of wonder and magic. They are not based on facts but on worlds of fantasy.