The National Forests are managed by the Federal Government not the states. There are a lot of equivalent state forests with campsites that are managed by states. This is true in Minnesota where I live. As for the differences, the National Parks are our treasures. They have prominent features. i.e. Yellowstone, Yosemite, Grand Canyon, etc. Forests are just that, forests. National Monuments like the Natural Bridges NM I mentioned are usually feature areas on a smaller scale though some can be bigger than some national parks. I suppose it is just management and what presidents and Congress acts on.
The Minnesota and Wisconsin national forests campgrounds list many as allowing only 22-26 foot RV lengths. I imagine they were originally set up for tent campers before RVs got more popular. Backing in, maneuvering and parking can be extremely tight. That eliminates larger Class C's and A's and trailers. There usually are no hookups but with some water available in the campground and some vault toilets. The National Forest Service has been improving some of the more popular CGs to accommodate larger RVs and even adding electrical hookups. But they are rare.
As I said, forests. National Forests usually don't have any amenities like hiking trails though many in the upper Midwest are located on lakes with a boat launch.