urban center vs metropolis

urban center and metropolis both are nouns.

Word	Noun	Adjective	Verb	Adverb
urban center	Yes	No	No	No
metropolis	Yes	No	No	No

As nouns, urban center and metropolis are synonyms defined as:

urban center and metropolis: a large and densely populated urban area; may include several independent administrative districts

Other synonyms of urban center include city.

urban center (noun)	metropolis (noun)
a large and densely populated urban area; may include several independent administrative districts	people living in a large densely populated municipality
	a large and densely populated urban area; may include several independent administrative districts