The fundamental divide between the British and Continental Europeans is a psychological one and an ancient one which has two sources.
Firstly, every continental European has a history behind them suffused with the memory of the Roman Empire. Peace, security and power is identified with a unified Europe roughly based on the borders of the Roman Empire and ruled by a version of Roman law. Ambitious Europeans have harked back to this again and again. Whether Charlemagne, or the Holy Roman Empire, or Napoleon, or the EU, this is an inherited and powerful memory of what security is.
By contrast we were the furthest outpost of the Roman Empire, and whilst integrated for 400 years retained many pre-Roman features including a mythology of resistance to imposed rule. The legend of Boadicea lingered (for some reason I hate the now dominant Boudicca spelling). Large parts of the British Isles were of course never conquered by the Romans. We were told to fend for ourselves as the Roman Empire collapsed, and some significant part of British consciousness was formed by that. Even before that point we had launched rebellious claims on the Roman ‘throne’ (such as that by Clodius Albinus in the 2nd century AD), and spent years separated from Roman rule under Carausius in the late third century AD.
Secondly, Continental European engagement with land empire in Europe itself was constant and unavoidable. Ours was by contrast limited to the period from the Norman Conquest through the Angevin Empire and was finally extinguished by the loss of Calais under Mary Tudor in 1558. Certainly this was a significant period of time, but our greatest success and power would come after it. Our ambitions became maritime and global, not landlocked and European.
Europeans are quick to point to the British Empire and its loss as an explanation of supposed British pigheadedness and refusal to share the European dream. They call it British exceptionalism and imply that our own recognition of our difference is somehow arrogant, ignorant or racist. They ignore the fact that their own psychology is shaped by a far older imperial project, that of Rome, one which was defined more by conquest than trade, by conformity rather than adventure, by static mental boundaries rather than by imaginative invention.
In every way the difference in this Roman experience had lasting effects, just as powerful on Continental European psychology and assumptions as Anglo-Saxon liberty was for British assumptions. Neither were ever actually looking for the same things whilst sharing membership of the EU.