The major difference between American English and British English that is immediately visible to most people is not something I consider grammar as much as... spelling. The two are related, of course, but I consider them separate components of language.
Anyway, I found a great website that enumerates the basic differences between American and British spelling rules. The most well-known change between languages seems to be the "(American) -or vs. (British) -our," or perhaps the "(A) -er vs. (B) -re," but there are many other variations that are more subtle or just unknown. I know I personally was taken by surprise when I found the word "meow" spelled (or spelt?) in the British version of one of the Harry Potter books as "miaow." And I read the LiveJournals of several Canadians; I had to have a double-take when one girl mentioned "foetuses" and "draughts" instead of "fetuses " and "drafts."
Of course, all those spelling variations still make sense; for example, "miaow" just elaborates, à la Eliza Doolittle, the vowel sounds of "meow."
But I still can't get over "gaol" (that's "jail" for my fellow Americans).