central city vs city center

central city and city center both are nouns.

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

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

Other synonyms of central city include city centre.

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