So: American: Past Simple - BrE Present Perfect.
Second: Have got vs. Have
Suppose you want to ask s1 if s\he posses a car:
1. Do you have a car?
2. Have you got a car?
And here is a difference. Both sentences are correct, but American English is rather like 1 (do you have) and British English "got" (have you got"
Easy-to-notice spelling differences. Eg. center (AmE) - centre (BrE), Different forms of irregular verbs: travel ->into past-> traveled (AmE) travelled (BrE). [AmericanE is a little bit more simplified)
Moreover: how do you call it- differences in naming things: American English - hood British English - bonnet AmE - trunk BrE - boot or truck vs. lorry
And spelling: Words ending in -or (American) -our (British) color, colour, humor, humour, flavor, flavour...
Words ending in -ize (American) -ise (British) recognize, recognise, patronize, patronise and so on and so fourth.
That is not all differences. But I don't have time to write about all of them right now