I was wondering about what's the difference because I always learned it as city centre, but now I read it like that, and... Is it the same?
"The City Centre" is idiomatically how most people refer to the main business and commercial part of a city - where most of the shops and businesses are. It may not be the exact centre of the city, geographically speaking (although it often is).
"A City's Centre" is not an idiomatic term like above, so arguably it could mean the exact centre of a city as it appears on a map; however I imagine that most people would assume you meant "the city centre" anyway. If your intention was to define the exact central point of a place you would need to be more specific and say something like "the exact centre of the city", or "the dead centre...".