At level language spoken, they are many different? How pronunciation?
British English and American English are just different accents of the same language. There are also different expressions. For example, American english speakers will say "pants" and for the british "pants" means "underpants" (or what Canadians say: "underwear"). The brits will say "trousers" for what Americans consider "pants". Most people will say that British English sounds more "proper", like the accent that the queen has when she speaks, and american english is what most people are used to due to the wide popularity of American movies... Here is a video I found of the different pronounciation of words https://www.youtube.com/watch?v=DKEM-juLxmM
Can you hear the difference?
The differences between British and American English are not as significant as people think. Yes, the pronunciation is different in certain ways (for example, the vowel sounds, and the pronunciation of the letter 'r'). And yes, a small percentage of the vocabulary is different, and there are certain expressions that are different too. However, the important thing to remember is that it is the SAME language, and we can always understand each other.
There are some vocabulary differences, too. Everyone mentioned the pronunciation, which is the most noticeable difference. Like they said, US regional differences are quite big too. Like Vanessa said, the vocab can be different as well. The pants and underpants example is a good one.
There are also big cultural differences that sometimes affect the language, for example conversational norms (saying 'have a nice day' or not) or humor. A comment that is meant to be funny in British English may sound mean in American English, or vice-versa. Check out this article for a lot of information on that. http://ideas.time.com/2011/11/09/the-difference-between-american-and-british-humour/
They are not very different.
I think the main reason we speak of "British" and "American" English is that for reasons of national cultural pride, Noah Webster (and others) in the United States decided that the U.S. ought to have its own set of standards. Thus, for example, people in the U.S. look to the Merriam-Webster Third International Dictionary as the authority for U.S. English while people in the U.K. look to the Oxford English Dictionary.
We (I'm a U.S. native) spell "color" without a "u" because Noah Webster thought it was a good idea.
If you looked only at how people actually speak, without being influenced by where the national boundaries are, I'm not sure that the difference between "general American" and British "received pronunciation" is any larger than regional differences within the U.S.
They're not very different. If a British person has a strong accent, it might be difficult for me to understand them but there are people from my own country who have accents that are difficult for me to understand. Because of television, music, movies and the internet, the differences are becoming smaller.