In Rugby actual physical contact occurs during scrums and tackles; whereas in football the front 7 usually has some form of contact on each play plus others at times. I suspect the number of tackles in rugby exceeds the number of tackles in football, plus most rugby tackles involve single players without a running start. It would be interesting to see if scrums produce the most concussions. In short, it's hard to have simple metrics.

So it is quite a bit more contact.

Rugby league has more data available than rugby union. League is the more violent of the two due to two rule differences. First, the defence must retreat 10 meters after each tackle which results in each run having a more velocity as both runner and tackler get a nice head start to build up speed. Second, you only have five tackles to go the length of the field - no first downs. So on every fifth phase, there is a kick and mad chase which results in some heavy collisions. Rugby union has less of these events.

Just last week there were two incidents in NRL where guys got hit so hard their hands were still in the air in a seizure position while they were knocked out on the ground. It’s not even head on head contact that does it. In one instance the player bent down for the ball and the defender’s hip side swiped his neck. In the other instance, the player was knocked backwards and the back of his head hit the ground.