As I understand it, Rugby has a lower incident rate of concussion, and it's more the outcome of accidents than as a direct aspect of the tackling. That said, it's a very different style of play in Rugby to American Football.

Rugby players just don't deliberately go in with their head for the tackle, which is something you often see with NFL players (some NFL teams have experimented with helmet free tackling practice and seen it improve the game-time tackling effectiveness)

The lack of a stop of play after every tackle in Rugby also tends to ensure players are opting for tackles that they can recover from.

In American Football the expectation is for contact / hit with every play. Contact is just a staple part of every single play for every player. In Rugby, contact is largely down to those with the ball, and those specifically attempting to tackle them. It's a very different contact profile, even before you consider things like helmets and padding and what that does to the mindset of the players.