state capital vs urban center

state capital and urban center both are nouns.

Word	Noun	Adjective	Verb	Adverb
state capital	Yes	No	No	No
urban center	Yes	No	No	No

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

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

Other hypernyms of state capital include capital, city, metropolis.

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