Just to correct any misconception that "soccer" is an exclusively AmE term ... it's not! In the UK (where it was invented), there are several kinds of football, the two main ones being Association Football and Rugby Football. I played both when I was at school in England, and they were always referred to as soccer (from the abbreviated form Assoc. Football) and rugger. The word "soccer" was coined in England in the 1880s, and was very widely used in BrE for over a century. Soccer was played with a soccer ball, rugger with a rugger ball (or rugby ball). By the way, the A in FIFA is for the adjective Association.
As association football became more and more popular in countries where it was the only kind of football, and so was called just "football" (or the local-language equivalent), and as more and more international matches and tournaments were held, "football" became the standard international word for the game, and so became increasingly used in UK English. Nonetheless, the only simple and convenient way of distinguishing between the balls used in (association) football and rugby football is to call them soccer balls and rugby balls.
When association football started to be played in the US, they couldn't call the game just "football"", because that meant American football (a derivative of rugby), so the British word soccer came in very handy. So they play (American) football with a football, and soccer with a soccer ball.
Next week I'm going to a soccer match, even if my younger fellow-countrymen might mistakenly think that's rather American of me.
In Ireland we have Gaelic football, rugby and football / soccer. We sometimes say soccer if we need to differentiate between Gaelic and football, the ball is round in both games and is called a football, maybe sometimes a soccer ball but not as often as football, because Gaelic is more popular than soccer / football. We get around this by calling Gaelic football Gaelic. A rugby ball is called a rugby ball and is oblong. Clear?