In American Football the tackler uses his body more as a missile, typically leading with the head. In Rugby the tackler leads more with the shoulder, making sure the head is out of the way, and it's more of a "wrap-up" technique.

Also, I would think that the velocities and masses involved in American Football are higher - it is more of an anaerobic sport due to the short play duration, frequent breaks between plays, and frequent player substitutions.