George III and Elizabeth II have roughkly the same power as a matter of constitutional law.
As a matter of constitutional convention, George had much more power.
Neither has the power to pass laws independently. Laws are made by Parlaimanet. However, in George’s time, he had more control over who got elected to the Commons. As a great landowner, pre-Reform Bill, the King would be able to choose or influence the choice of a large number of MPs - estimates range from 75 to 125 MPs.
There was also a sense that the King should have the people in government that he wanted, so many MPs, especially country MPs, tended to support the government the King appointed.
Third, the general sense was that the King had a major say in foreign affairs, which included the colonies.
The result was that there was the Prime Minister, but the King had considerable influence in the policies of the Government.
In practice, that meant that although there was considerable sympathy for the America colonies among some MPs (Burke, and Chatham in the Lords), the King had a major impact on the policy towards the colonies. Lord North, the PM, likely was able to stay in power mainly because he had the King’s support, until the British forces were finally defeated.
It’s completely different today. The Queen has much the same legal powers, but by constitutional convention (which is just as much part of the British constitution as the laws), she only exercises those powers on the advice of the Prime Minister. The PM, in turn, becomes PM because he/she has the confidence of a majority of the MPs in the Commons. Unlike George, Her Majesty has no power to influence the make-up of the Commons.