In the United States, we have areas designated as National Parks, and areas designated as National Monuments.
What are the fundamental differences between the two?
The difference is mostly one of administrative/bureaucratic elements. Both are protected areas, the difference is in who creates and manages them.
All National Parks are run by the National Park Service (NPS), part of the US Department of the Interior. Creation of new national parks is done through an act of congress. Creation of a new park using land administered by a different dept, such as a National Forest which is managed by the Dept of Agriculture, would also require congressional action to reassign that section of land.
National Monuments on the other hand, can be created on any federally owned land, through presidential authority alone. Congress can create monuments as well, but authority is not exclusive to them as is for NPS. Additionally, management of a National Monument can be done by whatever agency is appropriate. Many National Monuments are run and managed by the NPS, but not all. Some are maintaind by the US Forest Service, some by the BLM, one by the Dept of Energy, one by the Air Force and others. Whereas without exception all National Parks must be managed by the NPS.
Apart from that, there is really no significant necessary difference, as whether a given place feels like a park, museum, or something else will depend more on what the protected area contains rather than who's running it. Being a park does have a somewhat narrower criteria than national monuments, but you'll see more variation within the two groups than between.