These two major versions of the English language (Australians and others have their own) have been in competition with each other almost from the time the original 13 colonies were established. Bill Bryson notes in his book The Mother Tongue that Samuel Johnson (1709-1784) criticized American usage of words like glee, jeopardy, smolder and antagonize. What Johnson failed to realize is that these words had actually originated in England but had, by the 18th century, fallen out of favor there, while Americans retained them.
There are a number of other words and phrases coined in England that the English forgot but Americans remembered and that the British then thought sounded funny. Among those listed by Bryson are hog, mayhem, magnetic, chore, skillet, homespun, deck of cards (the English call it a pack of cards), progress (as a verb), platter (a large dish) and fall (as a synonym for autumn. Thanks to American books and movies, the British are now reclaiming some of these words as their own, as well as picking up a number of actual Americanisms.
Meanwhile, the British have contributed words to American vocabularies: miniskirt, radar, gadget, weekend and even smog. Americans might think they are the ones who invented these.