Money, my boy. War would jeopardize the profits.
So long as the British get to enjoy trade with America, they get all the profits the Empire craves, with none of the cost and hassle of say, invading and occupying a country many times larger than Britain itself, with a well-armed hostile population that, unlike the Redcoats, don't need to be shipped around the world or pulled from some other restive colony.
Lets assume for a moment the love of money and trade goods is gone, and the Brits get the dumb idea to try and conquer the USA. They would lose, and here is why:
The British Empire was mighty, but it had four distinctive disadvantages going into any conflict with the USA:
1) The Brits were a sea power, not a land power, and never possessed a massive well-trained Army to the point they could solve a problem by just throwing soldiers at the problem til it was subjugated and went away.
2) The few places where the Brits COULD use soldier's might, India, Africa, China, etc, they enjoyed a massive technological and industrial parity over them. "We have the maxim gun, and they do not." There was no such parity with the USA, and any war with the USA would be like a war with another European power, which the Brits avoided unless they could gather a coalition alongside them, to make up for the manpower disadvantage, as per above.
The Brits were never going to get a coalition against the USA, especially when the few powers that might be able to contribute enough men and ships to this war to make a difference, such as France and Russia, all enjoyed very warm relations with the Americans, and saw supporting them as a way to spite the British.
3) So the Brits go it alone - and here we have the next massive disadvantage of the British would face, logistics. This is where a massive globe spanning empire on which the sun never sets comes to be a burden - what troops the British had to throw against the Americans would have to be pulled from somewhere else that needed them far more, such as India. The Brits could not and would not be able to throw the full might of the British Empire behind a war with the USA, because to do so would jeopardize the Empire.
America meanwhile, might not be able to say, land the US Marines in London, but they most assuredly could defend their own borders, and likely make a serious go at conquering Canada. Unlike the British, who had a global empire to occupy, manage, maintain and defend, the Americans could and would be able to make defending the USA and beating back the Brits their sole priority.
This is why any conflict past a certain point - say, 1820 or so - the war on land goes to the USA, while the Brits rule the waves. A wolf against a whale.
4) The USA is growing every year in terms of population, industry, military might, resources, etc, and every year that goes by is another year the Americans come closer not just to being able to fight off the British, but to strike back, and to make it hurt.
UK vs USA in 1812 ended in a stalemate.
UK vs USA in 1848 ends with the USA taking most of, if not all of Canada. Especially since events elsewhere (Rebellions in India, Russia eyeing the Balkans) mean eventually the Brits would need to seek peace.
UK vs USA in 1896 ends with the USA likely taking anything in the Western Hemisphere flying the Union Jack, and maybe even dealing a wound to the British that would make the Boer War look tame.
UK vs USA anytime post 1918 ends with the USA destroying the British.
Short of absolutely everything going wrong for the Americans - we're talking near ASB levels - there comes a point where the British Empire would rip out it's own guts trying to take on the USA 1v1.