We stay in both, and some private parks are high on my list, like Trailer Ranch in Santa Fe for its easy city access....however,
We love state, National Park, county and other public campgrounds because they often have more scenic features, trees and space. You are likely to have a fire ring or grill, and sometimes find lakes, rivers and streams.
Many public parks have online reservation systems, so I can book any hour of the day to suit my schedule and have something in hand confirming my reservation, rather than having to call repeatedly (some private parks make it hard to reserve, oddly enough.)
Often I get to review available open sites online and book the site that meets my needs. Private parks rarely offer this chance.
Some public ones are actually more expensive than local private parks, but are better, so worth it. Yes, you do have to pay in full up front at most public parks, but I don't mind that.
But especially I like the ranger talks, nature walks, visitor centers with exhibits, films, and so on. You are given an opportunity to learn of the history, flora, fauna, geology, etc. of the place you are visiting.
To give just one example, this summer we followed part of the Lewis & Clark National Historic Trail, and stayed in Onawa, IA by the water at the L & C State Park where the Corps of Discovery had camped in 1804, and saw a replica of their keelboat in the water.