As far as I know there has never been a difference between Britain and Great Britain. There are various theories as to why the Great was added, but as far as I am concerned, there is one (and only one) plausible answer. Historically (and still in some languages, such as French) the same word was used for 'Britain' and 'Brittany', the NW corner of France that was colonized by Britons in the first millennium and thus became 'Little Britain'. Thus 'Great' was used to distinguish the old 'Britain'. Indeed the 'Great' is still essential in French. If you leave it out and say Bretagne instead of Grande Bretagne they will understand 'Brittany'.
Another reason that we are stuck with the Great is that GB is the only accepted abbreviation of Britain. It is used in the international abbreviation of our money (GBP), the name of our Olympic team, the stickers that are put on cars when abroad to show where they are from etc. In each of these cases there is a historical reason why the term is GB rather than UK. Northern Ireland has/had a different car registration system, and they are allowed to compete in the British or Irish Olympic teams. Our money was technically separate from Irish money until (I think) the 20th century, even if it was equivalent in practice towards the end.
I suspect most people in GB have an understanding that is partial at best, whereas those in both parts of Ireland are all too well aware of the difference.
To misconstruct what you said, they were the same – until 1/1/1801. I think they only started using the term UK when it became distinct from GB by the addition of Ireland. This is clearly shown in this Ngram, which also shows that Great was added to the front of Britain about half the time for many centuries, suddenly started a decline in popularity in the mid-20th century. I think this is because it is seen as too egotistical.
There has been a further problem since Brexit, which has partially split Northern Ireland from Britain, in a totally unsatisfactory and unresolved way. I usually hear the term Great Britain in this context these days.
On top of all these issues, there is no adjectival form for either Great Britain or the United Kingdom, with the term British being used for both, with the resultant possibility of confusion.