They're technically still different: Britain is the entire island, which contains England, Wales, and Scotland.

Britainia was the Roman name for the island. When Roman government in England collapsed in the fifth century CE, the island was politically fragmented, and the region we call England today (ie, not Wales or Scotland) was slowly unified by the kings of Wessex (south-west England). They identified themselves as descendants of the Angles and Saxons from Germany, and the name Angle-land, England, stuck to describe the region they controlled by the 10th century.

But Wales and Scotland remained politically separate, and were only united to England after half a millennium of war and politics (Wales being conquered finally in the 13th century, and Scotland conquered and lost in the 13/14th, and only unified to England when James VI of Scotland took over the English throne following the death of Elizabeth I at the beginning of the 17th century).

The 'United Kingdom' refers to the union of these different parts into a single country (plus northern Ireland, which is part of the UK, but not part of the island of Britain). But England still means specifically the region east of Wales and south of Scotland, not the entire island of Britain nor the modern political state that is the United Kingdom. It gets even more complicated when you consider that Scotland and Wales now have some measure of self government while still being under control of parliament in Westminster.

So the queen is queen of England, Wales, Scotland, and N. Ireland. The American colonists were English, and Scots-Irish, and from other parts of Europe; collectively, the colonies were under British rule. The redcoats were British because they weren't just from England - some regiments were recruited from other parts of Britain.

The terms are only synonomous among Americans because our geographical and historical knowledge is patchy, and we don't always realize - or remember - to make these distinctions (myself included - living 3000 miles away from debates about Scotish independence, it's easy to slip up and forget that England is not at all the same as Britain).

Officially, the country of England became part of the country of Great Britian in 1707 when the Act of Union between England and Scotland was passed in both the English and Scottish parliaments. (The two countries had shared the same monarch since The Union of the Crowns in 1603.)

American colonists considered themselves Englishmen for many reasons, but legally, all the 13 colonies except Georgia were established as (or became, through conquest, as per: New York, New Jersey, Delaware) English Colonies (only Georgia was established after the creation of Great Britain) so it is natural that they thought of themselves as English.

In 1800, another act of Union (this time between Great Britain and Ireland) subsumed both into the new country of The United Kingdom of Great Britain and Ireland.

So, in the Revolution, the colonists revolted against Great Britain, but the United States fought the War of 1812 aginst the United Kingdom.

Since then, most of Ireland has become independent, so the country is now The United Kingdom of Great Britain and Northern Ireland.

A very early example of "Britain" being used officially by the royal court is James I and VI's proclamation of the Union Flag in 1606 which refers to "Subjects of this Isle and Kingdome of Great Britaine" and then "Subjects of South Britaine" and "Subjects of North Britaine". Subsequent proclamations on flag usage refer to "England or South Britaine" and "Scotland or North Britaine". Note that although the term "Kingdome of Great Britaine" England and Scotland remained separate kingdoms under a common monarch at this time and weren't politically united into a single state until 1707.

"South Britain" as a term for England didn't really catch on but "North Britain" / "North British" for Scotland and things appertaining to it can be seen into the 19th century, such as the North British Railway Company.

Just a minor note - Elizabeth isn't the Queen of England. She's the Queen of the United Kingdom of Great Britain and Northern Ireland [and the queen of all the other places she's queen of, but that's irrelevant to the discussion at hand]

While people might refer to England and Scotland as different countries, it is absolutely one monarchy, and no longer just a dynastic union (she's not the Queen of England and the Queen of Scotland - the monarchy she heads is that of Great Britain and Northern Ireland).