First of all, the way the words get pronounced, but sometimes also the way they get spelled might differ. EX: "color" (American) / "colour" (British) - just to name an easy one. Just my opinion, but for me it sounds like British is slower and maybe a tad old/gentleman-like, while American is faster and more slang-like. Though, there also exist various British slangs.

For one thing, there's a major change in accent. Most languages have different dialects, well you could say English has theirs too. And as Chrisa mentioned, the spelling of certain words can be different, especially when it comes to involving the letter "U". I find American slower and more drawn out, even slightly easier to understand as a British accent, depending on where you come from, can be quite difficult (Especially in northern parts with its cockney accent)