I won't say anything new but hopefully just elaborate. If you want the easy answer g oto the final paragraph. I'm writing this now with hindsight and my answer is a lot longer than I thought it would be....
The UK nowadays is England, Scotland, Wales and Northern Ireland.
Great Britain is England Scotland and Wales.
Think of it geographically and historically. Britain is an island. No physical boundaries truly make one country from one another. At the end of the tenth century England became one country, and much bigger and more populous than its neighbours. We bullied Wales, and totally swallowed it (Edward I - the 'bad guy' from Braveheart took it). Sorry any Welshmen but from then on Wales and England were, and still are (though your assembly goes some way to make a distinction) one country. Just to appease any welsh let's make the distinct between legal political entity and nationality now, the welsh are NOT English, but our laws are basically the same.
Ireland was a country of warring chieftains. This made easy pickings for the English, though for many many centuries the English only truly controlled the area around dublin, called the Pale. I believe and am too lazy to check now but it was around the time of the English Civil war, when Oliver Cromwell was in charge, that the English made any real attempt to truly own the whole island of Ireland, which he did BRUTALLY. The Irish don't hold a great opinion of Mr Cromwell to this day. Ireland was encorporated in 1801 and lost its own parliament. Until the 1920s Ireland was part of the UK until most of it seperated, after many civil revolts, leaving only 6 of the 9 counties of Ulster still under British rule - 'Northern Ireland'.
Despite Edward I's best efforts Scotland (see Braveheart - but don't take it too seriously) was never truly subjugated. Our Union began when James VI of Scotland became England's king, James I, after the death of Elizabeth I. He was chosen not only due to his bloodline but because he was Protestant, like the English (kind of). But it was another 100 or so years before political union between England and Scotland, and 200 years before the union of them with ireland. In the mean time the King held the following titles... The King of England, The King of Scotland, The king of Ireland and even still, though hopelessly, The King of France. They were ALL seperate countries, they just happened to have the same king. This meant he was king of 3 (and a half - Wales) countries, not including France, all with different nationalities and 3 different religions.
Then it gets complicated. Scotland was isolated, being so far up north, and it had previously relied on the 'auld alliance' with france to stop the English from dominating them. However now we only had one king and the English bullied scotland and prevented it from flourishing. I'll add at this point I'm English, and trying to be impartial! The scots invested unwisely in a new colony in Panama at the end of the 17th century and became bankrupt. England only promised to bail them out on one condition - we'd own their asses. A very unfortunate episode in Scottish history that led the the formation of Great Britain, codified in 1707. Britain was no longer an island but a political entity under English domination. Many Scottish nationalists will point out at this point that the jacobite wars, mainly Bonnie prince Charlie, were wars to remove the English from Scotland but in fact these were religious wars, Catholic vs Protestant, and the majority of the Scots ( mainly lowlanders) fought on the 'English', ie Protestant, side. However 'inferior' in the Union they may have been the scots went on to become a major part of Great Britain, with major industrial centres, many famous generals and governors in the days of Empire and to this day many prime ministers, including (maybe debatebly Tony Blair but most certainly Gordon Brown).
The Irish were fully incorporated after some 'naughty' behaviour in 1801. The official title of the entity was until the 20's The United Kingdom of Great Britain and Ireland, changing to the United Kingdom of Great Britain and Northern Ireland, officially one of the longest country names in the world!
So to finally to answer your question no, but like many issues in life, it's complicated. Under 'nationality' on an important document the English will write British whereas the Scottish will write scottish. Foreigners will refer to the UK people, or the British as English. This is no different from the English speakers calling the Netherlands 'Holland' - it's just a simplified recognition of who the big boy on the block is in any kind of union.