If you're asked for your nationality, the correct answer (according to 'officialdom') is 'British' (or 'British citizen' which, of course, is not the same as 'British subject'!). Even more pedantically, you could write 'citizen of the United Kingdom of Great Britain and Northern Ireland'.
However the UK is (as defined by the UN) a 'nation'. While that term is often used as if it's synonymous with 'country', it makes sense (here in the UK) to reserve the term 'country' for the constitute parts of the UK. (Annoyingly, my brain told me to type 'constituent nations', instead of 'constituent parts' there, but that wouldn't make any sense at all!).
So if you're asked which country you were born in, the correct answer might well be 'England', rather than 'the UK'. However I'm prepared to bet that you could write 'England' on some forms, only to see it crossed out and replaced with 'UK'!
However 'Great Britain' (which, of course doesn't include Northern Ireland) is unlikely to be the required answer. The person asking the question will either require a 'precise' answer (i.e. 'England') or a 'broad' one (i.e. 'UK'). With exception of a few sporting events, 'Great Britain' isn't really recognised as either a 'nation' or as a single 'country' (since it comprises three of them), yet 'British' is recognised as a nationality!!!
Basically it comes down to the fact that our constitution (which doesn't even exist in a written form) has been cobbled together over many centuries, resulting in many anachronisms.
Chris
PS: I've had trouble trying to explain this, in any comprehensible form, to someone whom I assume hails from the same country/nation/state as me. I once tried explaining it to an American, with no success whatsoever ;-)