They are almost exactly the same as written. Differences in spelling are few and include words such as "Color" in the USA and "Colour" in Britain. Also, famously, Noah Webster when he made the Webster's dictionary of American English affirmed the American spelling of words such as "analyze" instead of the British "analyse". As spoken, they can be quite different in the accents, colloquialisms and slang that are used.

There are in common some differences between both of them.As you aren't an English native speaker and wanna learn English. You should determine what to learn according to your purpose of learning English whatever for travelling or studding or even as a knowledge.

There are many lists on the Net with British vs American words. Here is one of them: http://www.studyenglishtoday.net/british-american-english.html I've worked with people from the U.K. Schotland, U.S., Canada, New Zealand, Australia and South Africa and they all have a different accents and use of vocab. So even though they all speak English, there is a great difference and variety in vocab and speech. People are talking about maitaining one form of English now. One, which everyone can understand. Maybe one of the native English teachers knows the word for this langage?

I agree with IMK. The grammar is almost identical. I think it was Winston Churchill who said that "Britain and America were two countries separated by a common language"