county seat vs county courthouse

county seat and county courthouse both are nouns.

Word	Noun	Adjective	Verb	Adverb
county seat	Yes	No	No	No
county courthouse	Yes	No	No	No

As nouns, county seat and county courthouse are synonyms defined as:

county seat and county courthouse: the town or city that is the seat of government for a county

county seat (noun)	county courthouse (noun)
the town or city that is the seat of government for a county	the town or city that is the seat of government for a county