ten-gallon hat vs cowboy hat

ten-gallon hat and cowboy hat both are nouns.

Word	Noun	Adjective	Verb	Adverb
ten-gallon hat	Yes	No	No	No
cowboy hat	Yes	No	No	No

As nouns, ten-gallon hat and cowboy hat are synonyms defined as:

ten-gallon hat and cowboy hat: a hat with a wide brim and a soft crown; worn by American ranch hands

ten-gallon hat (noun)	cowboy hat (noun)
a hat with a wide brim and a soft crown; worn by American ranch hands	a hat with a wide brim and a soft crown; worn by American ranch hands