They are very close, but I think there is a subtle difference. "It's amazing that people collect so much stuff over the years." This is a very clear statement with you expressing this fact forcefully as if there is no question it is amazing and a fact that people do collect a lot of stuff.
"It's amazing how people collect so much stuff over the years." By adding "how" you express some slight confusion or uncertainty about how they do collect so much stuff. You are not questioning that they do collect it, just that you don't quite understand how or why it actually happens. Again this is a slight- very slight- amount of confusion. It softens the statement so it doesn't sound quite so forceful. If you changed people to "your mother" or some individual, then the statement would seem much more disapproving and even rude, where the one with how in it is much more polite.