American and British English are both "variants" of the English Language, more similar than different, especially with "educated" or "scientific" English. Most divergence is due to differences in national history and national cultural development, varying regional and local idioms and colloquialisms, and media/advertising influences.
The following general categories of difference between standard American English (SAE) and standard British English (SBE) each have their own sociolectic value:
I. Different Pronunciation, although same spelling
Advertisement (advert, ad)
Controversy, Laboratory, Secretary
Leisure, schedule, dance, dynasty
Renaissance, oregano, clerk [bank, office], ate
'PC'-influence examples: harass, harassment, Uranus, etc.
II. Different Spelling, although same pronunciation
Colour — color, Centre — center
Cheque — check (noun form [bank]; verb "to check" the same)
Defence — defense (noun form), Licence (noun form) — license
Alright — all right; Manoeuvre — maneuver; tyre — tire
Ageing — aging; Whisky — whiskey (U.S. & Ireland)
Gaol — jail
III. Same Term, different but similar spelling and pronunciation
Aluminium — aluminum
Polythene — polyethylene
Maths — math (shortening of "mathematics")
Rise — raise (more money in salary, wages)
IV. Same Words, Different or Additional Meanings in One Variety
Pavement (GB) = Sidewalk (US); Pavement (US) = Road (GB)
I married a homely girl. The opening of our new play was a bomb!
Nigel and Trevor purchased 7-day Travelcard season tickets.
Evelyn took a coin out of [his/her] purse (GB=male, coin purse; US=female, handbag)
The committee tabled the motion (put it on the table).
We all had tea and biscuits. (cf Harry Potter, 'crumpets' vs 'English muffins')
The corn harvest was exceptional this year. (cf. "maize", "sweetcorn")
We needed a torch for the dark trail. (cf. flashlight, electric torch, flaming torch)
IBM made over a billion dollars last year. (cf "thousand million"; changing GB standards)
Trousers (GB) = Pants (US); Pants (US) = underwear (GB)
Jumper (GB) = Sweater (US); Jumper (US) = pinafore [dress] (GB)
Shorts (US) = underwear (GB); Vest (US) = underwear (GB)
Pekka was climbing in the lime tree (lehmus, cf linden, basswood, common lime)