The real factor that makes American football better than any other sport in my opinion, and the one that gets overlooked more often than not even by a lot of it's biggest fans, is the fact it is not only a violent contact sport but also a virtual chess match between two teams. Just glancing at it it's hard to pick that up. Strategy, coaching, preparation, offensive and defensive awareness, split second decision making, adjusting strategy mid game and continually trying to stay ahead of the other team doing the same thing, teamwork, trying to deceive the other team of what your are trying to do and disguising your plays, and memorization of everybody's position and everybody's individual talents probably plays an order of magnitude more importance in it than any other major sport in the world. Intelligence, savvy, and coaching can play much more in winning than in other sports where the shear dominance of physical talent and the talent of a few individuals (and sometimes only one) often plays the overriding factor and sometimes almost the only factor (not that it doesn't also play a large part in American football). Part of the reason that happens is because actions are stopped and plays are continually set up on both sides, and it isn't one fluid event after another.
When it comes down to it that where it shines above everything else for me, and where I wish people would understand more of it.
I also like the fact there are a very wide variety of player types that are needed to be successful, with vastly different ability sets. A basketball, hockey, rugby, or soccer player are pretty much 95% the same for what type of athlete they are and what they do and their jobs are much more similar to each other player. In American football a lineman is vastly different and has a vastly different job than a receiver has for example.
Rugby > American Football
They play longer and harder without having to stop every 30 seconds. They don't need padding. They are actual athletes. If you think NFL players are Gladiators, Rugby players are fucking Spartans.
By the nature of the game and it's fluid format Rugby players don't get hit as hard most of the time. It's much harder to line up that good hit, and it takes more luck for it to happen. Stopping the game and setting up actually allows for more vicious blows. It continually allows people to get into position and line their hits up.