So it actually helps to be aware of language history. Obviously American English comes from England, but why do the two sound so different? Well, one of them changed over time, and it's probably not the one you think.
The American English accent is actually what was spoken back in the 17th century and before in England, and the British English accent changed after the American colonies were well established.
In short, the [bɪn] or [bɛn] (bin) pronunciations were first, as attested in old dictionaries and poems (by comparing rhyming words). It wasn't until the mid 19th century that British English speakers began using the [bi:n] (bean) pronunciation in order to have it fit more with the spelling (From "English Spelling and Spelling Reform", Thomas R. Lounsbury (1909)).
I realize this is an old question, but I hope anyone else looking up the answer to this question finds this helpful!