Why do British and American English spell things differently?

The divergence is substantially the work of Noah Webster, the American lexicographer whose dictionaries in the early 19th century promoted simplified spellings. He dropped the u from colour and honour, reversed the ending of centre and theatre, cut the doubled l in traveller, and changed defence to defense. His motives were mixed and both are documented: he genuinely believed English spelling was needlessly irregular and should better reflect pronunciation, and he explicitly saw a distinct American orthography as an assertion of cultural independence from Britain. His dictionaries became standard in American schools, which locked the changes in.

Not all of his proposals survived, he pushed for tung, wimmen, and iland, none of which took, so the surviving set is the portion that happened to catch. Meanwhile Britain standardized around Samuel Johnson's earlier dictionary, which preserved more French and Latin-derived forms. Some differences predate both men or arose separately, and a few common assumptions are wrong: the ise and ize endings are not a clean British-American split, since Oxford style has long used ize in British English. The practical guidance for writers is simply consistency within a document and awareness of your audience, since both systems are correct and spellcheck defaults will happily mix them if you let it.