There's a well put together article on Wikipedia about the
differences between the two dialects, and another decent one on
spelling differences.
From around 1600, the English colonization of North America resulted in the creation of a distinct American variety of English. Some English pronunciations and words "froze" when they reached America.
In many ways, American English is more like the English of Shakespeare than modern British English is.
Some expressions that the British call "Americanisms" are in fact original British expressions that were preserved in the colonies while lost for a time in Britain (for example trash for rubbish, loan as a verb instead of lend, and fall for autumn; another example, frame-up, was re-imported into Britain through Hollywood gangster movies).
Spanish also had an influence on American English (and subsequently British English), with words like canyon, ranch, stampede and vigilante being examples of Spanish words that entered English through the settlement of the American West. French words (through Louisiana) and West African words (through the slave trade) also influenced American English (and so, to an extent, British English).
😉
I'm all for calling the Americans lazy (
😛 ), however I'd have to say that, in my view, American English retains more similarities to "Modern English" (circa 1800), than British English does - which brings us to the logical conclusion that American English is more established than British, in their most modern forms.