inner city vs urban center

inner city and urban center both are nouns.

Word	Noun	Adjective	Verb	Adverb
inner city	Yes	No	No	No
urban center	Yes	No	No	No

inner city (noun)	urban center (noun)
the older and more populated and (usually) poorer central section of a city	a large and densely populated urban area; may include several independent administrative districts