Which should I use and why?
-- It seems that Tweetmeme is now recommending the Twitter Tweet button. But I still see lots and lots of Tweetmeme buttons around the 'net.
Use the official one.
Twitter took over the basic button last year, anyway. Any actual Tweetmeme buttons you still see were most likely just never updated/switched and there'd eventually just be a redirect of some kind on their end that leads to essentially the same result.