Independent cities are mostly relegated to Virginia (38 of the 41 in the nation), where they are simply a different sort of local jurisdiction. The Virginia Constitution allows that cities incorporated at the time of the promulgation of the constitution in 1871 (post Civil War) are independent of other local jurisdictions - namely the county. The US census bureau simply declares them to be county equivalents - i.e. the lowest, non devolved form of local government. In Virginia, what most people in other places would could call a "city" (that is, a devolved, sub-county jurisdiction, incorporated, and usually more urbanized than the surrounding county land) is called a "town". A set of examples can be found in the Northern Virginia suburbs of DC. The county of Arlington is directly next to the City of Alexandria, both of which are made up of the portion of Virginia originally ceded to the District of Columbia and retroceded back to Virginia later and both are nearly surrounded on the Virginia side by the County of Fairfax (the largest local jurisdiction in the state by population). Having lived parts of my life in all three, I can tell you that there is no fundamental difference for everyday life between the three unless you live in one of the towns located within Fairfax County (such as the incorporated Town of Vienna), at which case there is a second level of local government you have to deal with. In Alexandria, there is a mayor/council form of government. In Arlington and Fairfax both have County Boards, with the chair acting as the chief executive.

Keep in mind that this is in a highly urbanized area (several million people live in the three counties/cities in the DC suburbs). The actual effect of living in one of Virginia's independent cities might be more heavily felt if you live in one of the many rural independent cities, but I doubt it. The primary difference I can imagine is that with a city, it's a smaller jurisdiction than a county, and so individuals living there might have marginally more influence as compared to the county next door, but that is completely dependent on how the city is managed and its relative size to the surrounding counties.

As for the non-Virginia independent cities (Baltiimore, MD; Carson City, NV; and St. Louis, MO) I don't have a clue. My guess is that such independent cities provide similar benefits as consolidated city-counties, like Nashvville, TN or Louisville, KY, in which the city and county have been merged into a single jurisdiction, instead of the city being a sub-jurisdiction of the county.