99% of the time British and Americans will understand each other. With the widespread exportation of American films and series, British people are probably more likely to understand different accents etc, however Americans can struggle with regional accents from the UK.

Native speakers of English always understate the differences between American and British English. We don’t need to separately study the other variants to understand them because we have decades of exposure to each other’s speech in the media. This ability to understand without making a deliberate effort makes us jump to the false conclusion that they are ‘close’.

It is the same language. The differences are often exaggerated. We understand each others' written language perfectly. I'm a US speaker. To me, British English is not really any different than any other "regional accent." When I listen to the BBC World Service I recognize a "British accent." When I listen to old videos of US president John F. Kennedy I can instantly recognize a "Boston accent." If a British speaker talks about the "boot" of a car, I might be puzzled because we call it the "trunk." And if someone from New Orleans in the US talks about a "neutral ground" in a city street, I might be puzzled because around here we call it a "median strip." The difference is exaggerated because of history and national pride. We separated from England in 1776. We were a literate nation, with our own printers and publishers. A man named Noah Webster decided there would be a market for "An American Dictionary of the English Language." The fact that there are US and British dictionaries makes it seem as if the differences are larger than they are. I'm afraid to post links but Wikipedia has an excellent article on "Comparison of US and British English." Here's another indication that there's not much difference. There's only one English Wikipedia. It's a mix. Some articles are in US English, some are in British English. Few people ever notice. Search on "gramophone" and it shows you an article on the "phonograph." Search on "aluminum" and it shows you an article on "aluminium."

Forgot to mention, the spellings of some words are different as well :)

The pronounciation and some expressions are different between British and American English. You can watch some news reports from BBC and CNN, and you can know the difference between the pronounciation. As for the expressions, I would like to give you some examples: ground floor(BrE) and first floor (AmE).