headgear vs hat

headgear and hat both are nouns.

Word	Noun	Adjective	Verb	Adverb
headgear	Yes	No	No	No
hat	Yes	No	Yes	No

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

headgear: clothing for the head
hat: headdress that protects the head from bad weather; has shaped crown and usually a brim

headgear (noun)	hat (noun)
clothing for the head	an informal term for a person's role
stable gear consisting of any part of a harness that fits about the horse's head	headdress that protects the head from bad weather; has shaped crown and usually a brim
the hoist at the pithead of a mine

headgear (verb)	hat (verb)
	put on or wear a hat
	furnish with a hat