It's funny that only after making my own strides into the wider world (namely, outside the UK) has become second nature, I've really begun to reflect on this: am I British or English?
To tell you the truth, I've always considered myself English. The first time that "British" was brought into my consciousness was at the age of five or so, sitting in the classroom, and being taught the constituent countries of the United Kingdom and their capital cities. I remember having trouble pronouncing "Edinburgh", and the teacher kept telling me it was "Edin-bruh", not "Edin-burra". Another fond memory I have is the teacher pointing out that the island of Great Britain looked like a silhouette of a witch riding a pig, an image I have never since been able to shake.
And even at five years old, I was very aware of the concept of national identity. My primary school was in a village just outside one of the only remaining American airbases in the UK, so we had a wonderful mix of people in our school. Up until that point, I think I only really knew about France, from holidays - I was mesmerised by their different language and the fact they drove on a different side of the road. And I was faintly aware of some upside-down land called Australia where kangaroos lived (it could have easily been a place from a storybook, though). But really, it was exposure to these American classmates that made me realise there was a whole other world out there, one so big and so far away that it was almost beyond comprehension. Everyone in the school proudly fell into one of the following categories: "English" (i.e. not British), "American", or "half-and-half". I had friends whose company I would enjoy for a year before their military fathers flew their families back to the States.
"British" sounds quite sterile to me; it isn't really disclosing any information about my cultural identity. It's only something I would use if I'm talking about my citizenship, or if I'm comparing two places on a broad, general scale (British/German), or if I'm describing something that features in all four countries of the UK (Tesco is a British supermarket). If your parents are, for example, respectively English and Welsh, I can see why you might call yourself British. But both of mine are English.
When describing where I'm from when I'm travelling, it's actually a lot easier to say I come from England anyway. "Je viens d'Angleterre" rolls off the tongue a lot better than "Je viens du Royaume-Uni" (damn you, French "r"), as does "Ich komme aus England" than "Ich komme aus dem Vereinigten Königreich" (although for some reason, the Germans tend to like referring to it more geographically than politically - Großbritannien). And even in English, "I come from Britain" feels incredibly awkward to say.
When I was working in Austria, I knew fellow assistants who were mainly American and English, and there was one Scottish girl. I remember one day, I was describing to an Austrian resident of my student hall what I'd done the previous evening - I'd spent it at the house of the Scottish girl. I said to the resident that I had been at a friend's place, 'who's British too'. And then I was quite surprised at myself after I had said it. Especially since I had met Scottish people who preferred not to be called British.
But it would have been weird in this situation to say she was Scottish, because what I wanted to express to my Austrian flatmate was the similarity we shared from having grown up in sort-of the same country and so it was something of a home comfort to hang out with her, as opposed to with Austrians, amongst whom I was considered "the outsider". Saying she was Scottish would have emphasised the difference between my friend and I.
I've found it's actually overwhelmingly non-British people who use the word "British" to describe someone or something. In its most annoying form, it's in the phrase "British accent", which completely washes over the richness of accents, dialects and even languages within England, let alone the ones in Scotland, Wales and Northern Ireland. Even worse is when people say "British accent", and then in the same breath, "Scottish accent" - it makes my skin crawl.
It can be important to make these distinctions because whilst we do have our common points, there are indeed differences. For example, because I'm English I don't get a "free" university education like I would if I were Scottish. Or the fact that Scotland and England have the same currency, but different money. Or the fact that you'd never see a bilingual sign in England the way you would in Wales.
So it's kind of imperalist to say that there's a British accent, because not only does the one you're most likely thinking of only refer to a certain form of English accent, but it's also an accent that usually has connotations to upper-middle and upper class life.
The UK is four distinct countries, and millennia of blood has been shed so that various peoples inhabiting the British Isles retain the right to maintain their identities. The English language has dominated the original Celtic languages of certain areas; these minority languages are such strong markers of people's heritage. Read this page about the language of Scotland just to get an idea - I learnt quite a few things from it.