What is the distinctive feature of a state that distinguishes it from smaller political divisions?

Answer 1

States are sovereign. Which is a simple way of labeling an incredibly subtle distinction between them and cities/counties.

Basically, states are not sub-units of the federal government, they are subordinate quasi-independent governments in their own right. By contrast, cities and counties are sub-units of the state, and not independent at all.

The bottom line is that states are granted certain rights by the Constitution that enable them to do things across the entire state, where sub-units of the state cannot do those things.