National Monuments don't necessarily have to be something man-made. So they don't have to be like the Statue of Liberty, but they can be something completely natural like the Colorado National Monument which is just a big canyon carved out of spectacular sandstone. National monuments are protected areas like national parks, but they don't need congressional approval to become monuments. It is still under the national park system. So my interagency national park pass let's me into the national monuments as well as national parks.
sorry if this explanation is a little convoluted. There isn't much difference in my opinion. Just the approval process and funding.