There are lots of subtle differences... American football tends to have faster players, and that extra speed tends to make being hit more dangerous. They used to play with just leather helmets up until the early 1950s. Since then, the game started getting a lot more in revenue, and teams (especially leagues) wanted to protect their investments in players and avoid lawsuits from their families, and pads started being required. It's also a ridiculously high-contact sport, even moreso than rugby from what I've seen.
Forward passing was implemented because the games were apparently boring when it was mostly just kicking and running back in the early 1900s (about 1906, actually). As far as I'm aware, there is essentially no forward passing in rugby.