I'm involved in airport management and what this comes down to is a difference in funding. There are really 3 main categories of civilian airports:
-Privately owned, Private use (privately funded)
-Privately owned, Public use (privately funded)
-Publicly owned, Public use (publicly funded)
There are lots of differences in regards to requirements for things like markings and permissions, etc depending on the type. Some states are now allowing privately owned, public use airports to get limited funding in order to remain open, provided they comply with grant restrictions.