I don't know much about rugby (of either or any kind) but I'm under the impression that it has some kind of a culture of mutual respect. That might mean the players tend to exercise greater and more conscious control in taking contact despite the sport being rough.

I'm also wondering if that's partially connected to exactly the fact that they don't wear protective equipment. In many cases it's of course definitely helpful to have protection, but since it can also make the risks less obvious, using heavy padding might also encourage heavier contact. If, on the other hand, none of the participants are under any illusion of safety, that could discourage recklessness and perhaps even encourage a sense of mutual respect.

Helmets generally don't offer that much protection from concussions. They do reduce the risks of other serious head injuries such as fractures, as well as more superficial injuries, but if helmets make people think they can take heavy contact with abandon, that might increase the number and severity of concussions.

My opinion: it's the padding on the other parts of the body (particularly the shoulders) that makes NFL so dangerous. If you ran directly into someone at full speed without padding you'd injure yourself as much as you'd injure them, whereas the pads NFL players wear allow them to use their bodies as guided missiles. Rugby collisions (generally) happen at lower speeds, where the intention is to wrap up and grapple the opponent to the ground, not blast them off their feet. (There are also on-side rules in rugby, and no blocking away from the ball, so many, many fewer blind-side hits occur.)

The stoppages in American football also mean bigger players with worse stamina are viable, since they get a break after even short plays.