The sport now known as American football or gridiron football changed drastically in the past century or two. When the game first gained popularity in North America, it started as rugby football and the ball was generally advanced by kicking it. Consequently, the term "football" had caught on during the sport's early days in North America.

It's also worth noting that were a bunch of different sports that had names which were variations of "_____ football" in the same manner as rugby football. One of those was association football, the full name for the sport known as soccer in the US and "regular" football elsewhere. Where did the term "soccer" come from? In the late 19th century, some slang was developed at Oxford University in England that informally used "_er" as a suffix at the end of words. For example, a five pound note would be called a "fiver" in this slang. These Englishmen applied this slang convention to the term association football, shortening it to assoccer. Over time, this got shortened from assoccer to just soccer. I'm not sure how, but this term made it from Europe to North America, where it ended up being adopted.