In summary, the differences between each of these protected lands are the motivations for protecting the land and for what usage. The National Park Service probably has the most straight forward mission: preserving natural and cultural resources for “enjoyment, education and inspiration”. Essentially conservation and recreation.
This means the Park Service makes an effort to make these lands accessible to the U.S. citizens. As a result, these areas are the most likely to have infrastructure such as roads, campgrounds, backcountry campsites, etc. and bring the majority of casual recreators to nature.
The National Forest Service mandate is much more difficult to figure out than the National Park Service since they use different land management strategies for different types of lands. However, the primary purpose of these areas is to protect these forests as a resource for the federal government. They are less worried about preservation than they are ecology health and as a result they can have more back roads and can even allow logging.
Recreation is allowed in these areas, but since their mandate isn’t necessarily recreation there typically are no facilities for campers or backpacking sites. This means backpacking these areas typically involves dispersed camping and as a bonus, typically doesn’t require a permit!
Section 4 in this report gives the most high-level, straight-forward details about the Forest Service.
National Monuments are a special classification of land and can be managed by various agencies. This classification was created to protect areas of historical or scientific significance which did not fall under the National Park Service’s mandate of recreational value. Originally the classification was created for protecting prehistoric Native American ruins found in the West.
As a result, you are less likely to find backpacking here. However, campgrounds are common and you’ll likely see some really cool historical or geological parts of the United States.
Further Reading at Wilderness.org
Demonstration Forests areas are similar to National Forests, except their role is as a proving ground for research and forest management / foresting techniques. As a bonus since they are owned by the government they are also open for similar recreation opportunities.
Managed by U.S. Fish and Wildlife Service, these areas meant to conserve not land, but wildlife resources for U.S. citizens. Since these are wildlife and not land specific, most public-use efforts revolve around preservation, education and where appropriate, harvesting these resources.
Nothing very special here. There is no real difference which level of government is involved in the land protection, other than who is paying for the upkeep.