Soccer was an abbreviation used by posh people. Associate football -> sociate -> soccer. Much like rugby is called ruggers by the same group of people today. It was an informal term.

Association football was popular amongst the working class in the UK, who didn't use the same types of abbreviations. So it wasn't referred to as soccer by the them. When radio/TV became common the presenters wouldn't use abbreviations like soccer and so it was referred to as Association Football or Football.

In the US the posh abbreviation took over, likely because many British travellers to the US would be posh and not working class. At least the ones traveling for leisure and taking part in sports activities. Working class would mostly be immigrants and wouldn't be brushing shoulders with those in sports media.

American call the rugby like sport, American Football because it is played on foot and not horse. It would also share a common ancestry of completely moving a ball from one place to another on foot, like football and rugby.