fool's cap vs hat

fool's cap and hat both are nouns.

Word	Noun	Adjective	Verb	Adverb
fool's cap	Yes	No	No	No
hat	Yes	No	Yes	No

As nouns, hat is a hypernym of fool's cap; that is, hat is a word with a broader meaning than fool's cap:

fool's cap: a cone-shaped paper hat formerly placed on the head of slow or lazy pupils
hat: headdress that protects the head from bad weather; has shaped crown and usually a brim

Other hypernyms of fool's cap include chapeau, lid.

fool's cap (noun)	hat (noun)
a cone-shaped paper hat formerly placed on the head of slow or lazy pupils	an informal term for a person's role
	headdress that protects the head from bad weather; has shaped crown and usually a brim

fool's cap (verb)	hat (verb)
	put on or wear a hat
	furnish with a hat