In American football you also may only tackle the ball carrier. You can initiate contact other guys (usually in the form of blocking), and that can be very violent at times...but I don't think there's a scenario where you can tackle someone without the ball (and it not be called for a penalty).
The main difference in tackling technique between American Football and both codes of Rugby is use of the arms. In Rugby the arms must be used to tackle the opposing player. A shoulder charge, with no arms used, is illegal and may result in expulsion from the game. In a recent New Zealand vs British and Irish Lions game, NZ player Sonny Bill Williams was given a red card and sent from the pitch for this exact offence.
The other main points to note are that the opposing player can only be tackled below shoulder level and, as you've pointed out, in open play only the ball carrying may be tackled (or blocked).
Tackling the ball itself and not the player.
I don't have stats to back this up, but my general impression over the last couple of seasons is that less scrums require resetting.
I've thought this may be why American football has more of the full body collision, where two (very large and powerfully built) men collide at high velocity almost horizontal to the earth. They're trying to gain as little as a few inches in the collision.
I've heard complaints from Australian rules players who come to the US about tackling - that the way US players have learned to tackle doesn't have much strategic value and greatly increases the possibility of injury, buy they continue to tackle this way when they join Australian rules leagues.
That's what I've heard and thought about, I do want to be clear that I don't know too much about all this - what I've written is more of a question in case someone in the know could weigh in.