Merriam-Webster's Learner's Dictionary
6 [count] : the main part of a town or city where there are many stores, restaurants, offices, etc.
▪ the town/city center
▪ efforts to rebuild the city's center [=downtown]
Do city center and city's center mean the same thing?
I'd have to think hard in order to consider a real discernable difference beyond that of a nuance between the two phrases.
Maybe someone else can talk about some sort of difference between the two.