city center vs central city

city center and central city both are nouns.

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

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

Other synonyms of city center include city centre.

city center (noun)	central city (noun)
the central part of a city	the central part of a city