story vs fairytale

story and fairytale both are nouns.

Word	Noun	Adjective	Verb	Adverb
story	Yes	No	No	No
fairytale	Yes	No	No	No

As nouns, fairytale is a hyponym of story; that is, fairytale is a word with a more specific, narrower meaning than story:

story: a trivial lie
fairytale: an interesting but highly implausible story; often told as an excuse

Other hyponyms of story include cock-and-bull story, fairy story, fairy tale, song and dance.

As nouns, fairytale is a hyponym of story; that is, fairytale is a word with a more specific, narrower meaning than story:

story: a message that tells the particulars of an act or occurrence or course of events; presented in writing or drama or cinema or as a radio or television program
fairytale: a story about fairies; told to amuse children

story (noun)	fairytale (noun)
a structure consisting of a room or set of rooms at a single position along a vertical scale	a story about fairies; told to amuse children
a piece of fiction that narrates a chain of related events	an interesting but highly implausible story; often told as an excuse
a short account of the news
a trivial lie
a message that tells the particulars of an act or occurrence or course of events; presented in writing or drama or cinema or as a radio or television program
a record or narrative description of past events