They are entirely separate concepts, in the US at least. Counties are a common way for a state to subdivide its territory, and are related to State government. While Cities tend to have an entirely separate governmental structure, subordinate to the State government, but still a separate local government.
Where I live, there is a Federal government and State government, of course, but also a County government. Then, I happen to live in a Village, which is also part of a Town, within the County.
So, I have no less than 5 entities I owe tax to. Six, if you count the School District separately. (Which I do, since it can also levy taxes). Fun!