From my admittedly not at all expert viewpoint, the main differences between rugby and American football are the degree of armouring, the specialisation of positions and the frequency of stoppages in play.
In my view it is the latter two that are more significant than the degree of armour allowed.
American football seems to stop play constantly. Entire teams are rotated on and off the pitch at various points depending on what is going on. Players exist to play a few seconds at a time, executing a limited number of tasks.
Play in rugby (to varying degrees depending on code) continues for longer periods in one go with significantly greater likelihood that say a prop forward may end up having to run around the pitch for 10-15 minutes without stopping other than to tackle someone, pick themselves up and do it again on the other side of the pitch.
Therefore all-round fitness is slightly more important than in American football where for example, the ability to exert maximum force over one or two metres of ground may be far more important than whether that person can run the length of the pitch at that speed or do anything other than block or tackle.
Every position in rugby may (and probably will be) called on to perform all of the skills in rugby (running, passing, tackling, kicking, gathering the ball from the ground, rucking, mauling, catching the high ball).
We’ll leave out the various technicalities of scrummaging and the line-out since league doesn’t have these.
League (much as it pains me to say it) tends to require more all-round skills from its players which is why there tends to be a much smaller degree of variation in body shapes in league than in union.
This tends to restrict player size and weight since the number of giants that can also run really fast for more than short periods, be agile and skilled with their hands is more limited (and they can just play basketball far more safely 
).
Despite this player sizes are increasing and so are collision speeds and therefore concussions and other serious injuries.
The armouring is more a consequence of the injury risks than a contributor. Players will happily bash into each other without armour - rugby proves that. They may be more prepared to do so if they think they will get away it thanks to padding, etc. but not much more so.
In short, rugby-style padding will not solve the concussion problem.
I don’t think there is a way to solve it short of entirely changing the games.
