city vs urban center

city and urban center both are nouns.

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

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

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

Other synonyms of city include metropolis.

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