gag vs funny story

gag and funny story both are nouns.

Word	Noun	Adjective	Verb	Adverb
gag	Yes	No	Yes	No
funny story	Yes	No	No	No

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

gag (noun)	funny story (noun)
restraint put into a person's mouth to prevent speaking or shouting	an account of an amusing incident (usually with a punch line)
a humorous anecdote or remark intended to provoke laughter

gag (verb)	funny story (verb)
struggle for breath; have insufficient oxygen intake
make an unsuccessful effort to vomit; strain to vomit
cause to retch or choke
make jokes or quips
be too tight; rub or press
tie a gag around someone's mouth in order to silence them
prevent from speaking out