Translation:We are from France, but we live in Great Britain.
Yes, it is, yes, these are not the same... and yet we just use "Wielka Brytania" almost all the time - probably only apart from serious political contexts, when this difference is really crucial. In such a sentence as this one, I would be surprised to hear someone say "mieszkamy w Zjednoczonym Królestwie".
In fact, I just checked Wikipedia (sure, not a perfect source, but still), and the English article on "United Kingdom" and the Polish one on "Wielka Brytania" are considered equivalent articles.
I guess this is kinda similar to using "America" for the United States - technically wrong, but used all the time.