National Parks run by National Park Service. Federally protected lands. There are lot fewer of them. They are more tourist-oriented and are treated like natural wonders for the public to experience. Very high restrictions to protect the land (staying on trail in certain areas, pets, campfires, leave no trace, camp and wilderness permits, manicured roads and trails). Has an entrance fee. Patrolled by park rangers. Often has crowds.
National Forest run by US Forest Service. Also federally protected and managed. There are a lot more of them and aren't marketed with much grandeur as a national park. They often contain a maze of rough, less-maintained forest roads. You can camp anywhere in them for free mostly without any fee or permit requirement, so it's sort of like wilderness. Less stringent rules of what you can and can't do. Very easy to drive into a national forest and see no one around. If I'm ever on the road, I'll sleep in a national forest or other public lands