It's not settled science but there's decent evidence that the padding and helmets contribute to the more violent nature of American football (where serious and long term injuries are more common than in rugby).
Another reason is the lack of laterals in American football. In rugby, if you commit too hard to tackling a player, he laterals the ball to the player next to him, and you're now in a really, really bad defensive position. In contrast, laterals are so rare in American football that you can commit as hard as you want without consequences.
Neither of these things go away if you get rid of the helmets and pads; it just makes it more likely that people will die on the field, as they frequently did before helmets and pads were introduced. At one point, football was killing so many college students that President Roosevelt threatened to ban it.