The short answer is that British and American English are very, very similar. I believe that the similarities are far greater than those separating Spanish & Catalan or Chinese dialects of many neighboring villages. The bottom line is that British and Americans understand each other without difficulty. To the untrained listener, our differing accents seem to be the only thing separating us. However, we have occasional misunderstandings of words and our formal writing skills will be lower outside of our own countries. Americans might joke that they "fixed the bugs" in English upon arriving in the New World, while the British might say that Americans must have picked up many bugs while on our journey at sea! For sure, Americans refined the language, but to a much lesser degree than done during the Old and Middle English evolutionary periods. In the process, American English lost some British vocabulary and formality. In fact, American English has added new meanings to existing British words and created distortions and errors that can hardly be called "refinements". And, American and British English have both independently introduced new words during the past several hundred years. With greater and faster communication across the world, we are seeing more cross-pollination between American and British English. In fact, as the world becomes an increasingly connected place, the baton for evolving the English language is being passed from the USA and UK to English speakers throughout the world. All of you will evolve the language while the rest of us will be trying to keep up! Traditionally, dictionaries such as Webster's in the USA and UK equivalents have standardized the English language. With English's emergence as a world language, I wonder whether such region-specific standards will eventually give way to an international standards group, just as former standards bodies like the USA’s ASA and Germany’s DIN were folded into the ISO.

This is a very, very common question. It's a pity we can't search past questions so easily anymore. The fact is, you can easily look up the answer yourself: http://en.wikipedia.org/wiki/Comparison_of_American_and_British_English