They do have a lot in common, but they are very different from each other. Not everyone lives in a city, but everyone lives in one of Utah's 29 counties. This is where we throw out those terms incorporated and unincorporated Salt Lake County.
means that you live in a portion of Salt Lake County that does not have a municipal (city) government. Your county mayor is your only mayor.
means that you live in and are represented by a local government AND county government. You might have a city mayor and a county mayor.
It is like going to the grocery store. You live in downtown Salt Lake City. Chances are you go to the small convenience store down the block. You can get most of you essential goods and services there but occasionally like to supplement the selection by going to the larger grocery store a few miles away.
If you live in unincorporated Salt Lake County, however, there is one stop shopping. You don't have a small convenience store down the block so you go to the big grocery store nearby. That is your only option and you can get everything you need there.
City government is the small, convenience store grocer. County government is the larger grocery store that is a little farther away. And state government? Well, that's the big warehouse club of governments. You go there to stock up on big items like education, highways and state parks. It is all good stuff, but it is so big it can be hard to navigate sometimes.