Well it is often said in parts of Europe that a language is just a dialect with an army, but I don't think that applies in this case.

The differences between British and American English are insignificant. Both dialects are entirely mutually intelligable, and due to the cross-pollination of cultures most speakers are familliar with local idioms and variations. I grew up in the UK with shows like Friends whilst many Americans watch Doctor Who or Dowton Abbey. We here different dialects all the time.

There's just as much variation between dialects within the UK or the US as there is between say London and New York. In Scotland for example 'yes' becomes 'aye' and 'small' becomes 'wee' in casual speech. Likewise the use of 'y'all' in southern speech as the second person plural pronoun is common in many Southern states, but not something you'd expect to encounter in New England.

Lastly, using the term "British language" would be incorrect, as Britain has never had a homogenous linguistic culture. English is evolved from the language of the Germanic tribes which first came England in the 5th Century, and mixed with the French spoken by the Normans who invaded in the 11th Century. Other parts of the British Isles retained their linguistic herritage and many of those languages are still spoken today, including Welsh, Irish, Gaelic, Scots and Cornish. As the English came to dominate the islands politically, English became the language of government and the other native languages were repressed. They are now enjoying a resurgence, thanks in part to passionate people who kept their language alive when they would have been punished for using it. Welsh in particular is widely spoken, and in Wales you will see most services and road signs are in both languages.

I freelance doing editing and writing for a US company but for the European market.  We use the terms ''UK English'' and "US English'' (or British/American, same) and for us it is very important which one we write in. Believe or not, the company pays people to do UK to US English ''translations'' (yes, I think this is silly).  But there are a lot of spelling, punctuation, grammar and word choice differences that are important in correct written UK or US English.  More than I ever realized before doing this job, even though I'd lived in the UK. 

HOWEVER, unless you are aiming to be a translator, the differences are so small that for English learners I just wouldn't worry about it.  We all understand each other.  Mostly. 

I don't think anyone would be offended by the use of those terms, though.  Even if most Americans think that they don't have an accent, and everyone else in the world does.  ;-)

They're far more similar than they are different. Under no definitions are they separate languages. They're 100% mutually intelligible in both speaking and writing, and children and English learners won't notice a difference between the two until they've had lots of exposure (as opposed to learning one, then 'breaking through' to the other, which doesn't happen). 

Also keep in mind that most American accents are descended from the accents of south west England.

Have a look at this: https://www.youtube.com/watch?v=PzxuDyNC1bQ

You can hear him switching between his native accent and a more 'standard' British accent. His native accent sounds like an American accent with a tinge of British.

Accents from the New England part of the US are descended from the lower class accents of south east England.

So you can see that it's not as simple as American English versus British English. American English is descended from at least two different varieties of English in English, and, by extension, American English and RP English (the closest thing to 'standard' British English) are closer to each other than other accents in the UK.

In the United States, we all say that we "speak English."

Noah Webster's first dictionary was entitled <em>A Compendious Dictionary of the English Language</em>, and the one that made him famous was entitled <em>An American Dictionary of the English Language</em>. The closest thing to an authority for U.S. English is <em>Webster's Third New International Dictionary of the English Language, Unabridged. </em>My favorite desk dictionary is <em>The American Heritage Dictionary of the English Language.</em>

It's really the same language. The differences between U.S. and British English are no greater than the differences between regional varieties of English within the U.S. There will be occasional moments of disorientation or misunderstanding between U.S. and British speakers, but there can be occasional moments of difficulty between U.S. speakers. The first time I heard someone from New Orleans use the phrase "neutral ground" I didn't know what he was talking about.

Furthermore, I think that British and U.S. English are actually becoming more similar as time goes by.

The more Spanish I learn, the more convinced I become that the difference between e.g. Mexican, Colombian, and Argentine Spanish are as wide as the (small) differences between U.S. and British English.

I agree with all the answers above. Emily is right; there definitely are those differences between British and American English, but, for the most part, those differences rarely ever hinder communication between speakers in the UK and the US. Well, the only thing that might hinder communication is that some Americans find a few British accents hard to understand. Other than that, if you're fluent in one, you're basically fluent in the other given that fluency is defined as "being able to express oneself easily and accurately."