What's interesting is that in Rugby you wear basically no protective gear, which is probably why it has less injuries. With a helmet and armour you feel much more invulnerable.
It is tackles above the shoulder (not waist) that are forbidden (along with striking an opponent with a stiff-arm whilst tackling, tripping, picking up and dropping a player etc.).
Fouls happen. I don't mean to diminish that, but when the base of the game is to disallow bad tackles then there will be far fewer incidents of bad tackling.
It's also that American football has a lot of downtime between the relatively short plays, whereas Rugby has a more constant style of play. An entire NFL game consists of roughly 11-12 minutes of actual gameplay -- the most recent numbers for rugby I could find put it at ~35 minutes. So, players are less fatigued and can hit harder and faster.
