jest vs fool around

jest is a noun but fool around is not a noun.

Word	Noun	Adjective	Verb	Adverb
jest	Yes	No	Yes	No
fool around	No	No	Yes	No

As verbs, fool around is a hyponym of jest; that is, fool around is a word with a more specific, narrower meaning than jest:

Other hyponyms of jest include gag, quip, arse around, fool, horse around, pun.

jest (noun)	fool around (noun)
activity characterized by good humor
a humorous anecdote or remark intended to provoke laughter

jest (verb)	fool around (verb)
act in a funny or teasing way	indulge in horseplay
tell a joke; speak humorously	commit adultery