I think this is true because football players wear a lot of padding, while with soccer they don't wear any...
The thing is, American Football is a collision sport. Association Football is a contact sport. The difference between the two is that a contact sport has limited contact (for what it's worth, I guess the NFL is a contact sport... if your name is Tom Brady).
You bring up padding... correct an American if he is wrong, but don't Rugby players wear limited padding? Perhaps around the same level of padding as Soccer players? Yet Soccer has 323% the frequency of injuries as Rugby, which is a collision sport, unlike Soccer.
Unfortunately, I can't go into further depth because there isn't much differentiation between the severity of various injuries (which is why I talked about frequency).