National parks are managed by the National Park Service (Department of the Interior) with a preservation mission: protect natural and historic resources “unimpaired for future generations.” Recreation is welcome, but rules are generally stricter—hunting is usually forbidden, dogs are often barred from trails, and motorized trail use is limited.
National forests are managed by the U.S. Forest Service (Department of Agriculture) under a multiple-use mandate. Besides recreation, forests may support timber, grazing, mineral uses, and a wider set of outdoor activities. Hunting is often allowed, dogs are typically permitted on trails, and some forests offer motorized trails.
Adjoining park and forest lands can look similar on a map but follow different regulations, so check which agency manages the ground under your feet.
Source: National Park or National Forest? — U.S. National Park Service (Great Smoky Mountains)