state capital vs city

state capital and city both are nouns.

Word	Noun	Adjective	Verb	Adverb
state capital	Yes	No	No	No
city	Yes	No	No	No

As nouns, city is a hypernym of state capital; that is, city is a word with a broader meaning than state capital:

state capital: the capital city of a political subdivision of a country
city: a large and densely populated urban area; may include several independent administrative districts

Other hypernyms of state capital include capital, metropolis, urban center.

state capital (noun)	city (noun)
the capital city of a political subdivision of a country	people living in a large densely populated municipality
	a large and densely populated urban area; may include several independent administrative districts
	an incorporated administrative district established by state charter