The term "British English" really only refers to writing. It doesn't make any sense in relation to pronunciation, though outsiders who use it that way are perhaps referring to a particular accent called Received Pronunciation, or modern accents resembling it such as Estuary English. It's hard to tell: the term means virtually nothing to Britons themselves.

British English can be distinguished from American English by the spelling: the familiar -re vs -er, -our vs -or, the variation in double consonants in words such as traveller, and the use of digraphs in words such as oedema. There are also differences in individual words, such as grey, gray.

In almost all cases the Southern Hemisphere Englishes align with British, though you do see spellings like color in Australia, even in newspapers (though never in books). Furthermore, in almost all cases, all parts of Britain use the same spelling: there is the odd exception such as gray being met with in Scotland. Ireland uses the same spellings as Britain. So what (North) Americans perceive as British ('Britishisms' as they sometimes call them) are more likely to be general non-American uses.

There are things that have names used in Britain but not elsewhere: lorry, which is truck in other Englishes.

No doubt there are quite a few popular phrases and colloquialisms that diffuse over Britain by the influence of television, but have not diffused overseas, and there may also be a few grammatical quirks that are widespread all over Britain and less so elsewhere. Possibly have and haven't where other Englishes use have got and either haven't got or don't have. (These also occur in Britain; the distribution is one of age and class: bare have is now quite conservative.)

But basically, the notion "British English" is almost entirely one of spelling, with a few cultural notions such as pelican crossing that occur across Britain and aren't used outside. In normal speech it's unlikely that a Briton would use such characteristic words enough to be noticeable.

There is no such thing as a British accent, nor British grammar, nor British slang.

The dialects and accents, and their corresponding treasure-hordes of slang and the like, vary within Britain much more than outside, because being the home of the language it has had vastly more time to accumulate variation.

Nor is there any such thing as an English accent, meaning "the" accent that an English person speaks with. This is not a matter of fine distinction, as e.g. distinguishing Australian from New Zealand, or Louisiana from Texas: the corresponding fine distinction in Britain would be between say Yorkshire and Lancashire.

We need to distinguish at least four major varieties (dialect groups rather than single dialects):

To this list may be added two others at an equal level, for comparison: If I had to make a crude division of English into dialect groups, that is the sixfold division I would make. The Southern Hemisphere varieties, South African and Australian/New Zealand, belong to what I am calling the Southern English dialect, though they have strikingly different accents.
Cletus the Foetus has suggested that American has some dialects that are as distinct as those within Britain: see American English. I don't know enough to comment in detail. The Newfoundland accent does seem to be striking different and perhaps should rate its own entry in the above list.