American Football is actually closer to rugby with extra padding than it is to Football (Soccer).
Back in 1863 in England, two football associations (association football and rugby football, aka Rugby Union) split off from each other. One of the reasons is that the Association people wanted to only use feet to move the ball, while the rugby people wanted to use both hands and feet (and bits of bone-smashing violence).
Association football had a nickname in Britain formed from the letter SOC in the word Association, which was the word “soccer.” Supposedly this happened at Eaton, where the game was also sometimes called “footer.” Rugby’s nickname is “rugger.” These are part of the “Oxford -er” coinages found in Britain, where the -er sound is added to a word, a bit of slang thought have been borrowed from the slang of Rugby School.
The first recorded use of “soccer” (sometimes spelled “socker”) was in 1895 and the term was made popular by a prominent English footballer, Charles Wreford-Brown (1866–1951).
This nickname (soccer) is often used today to refer to football in countries where another form of football exists, primarily the USA (although Australia has its Aussie rules football, or footie, and there is also Canadian Football, as well as burgeoning leagues in Italy).
In essence, the British both standardised the rules of the game and invented its nickname of “soccer”. As often happens, the colonies have kept on using an older term that has since been abandoned by Britain itself.