It's the same as ringette vs ice hockey in Canada.
Body contact is penalized, the puck is a rubber ring, and the sticks are straight without a paddle so you can't make a slap shot at the goalie. As a consequence, only little girls play it.
Football started off as a sport much like rugby in ancient Rome and Greece, and then in medieval times it transformed into "mob football" where again the ball was both kicked and carried by hand with the intent to score a goal by any means. They didn't call it football but just "playing ball". Football became distinct when the aristocratic kids started inventing new rules to only touch the ball by the feet, while the common people just played the game like it was rugby.
In other words, football is a dandier version of rugby made up by upper class people, while American football retains the original character.