Who has more power in the British government, the Queen or the Prime Minister?
The Queen's powers are purely formal. One way of describing her role is to say that she reigns, but doesn't rule. Nonetheless, she does have a very important role to play within the British constitution, providing some measure of continuity amid the ever-changing world of party politics. Hence the political system of the United Kingdom is referred to as a constitutional monarchy.
The government of the day is always officially known as "Her Majesty's Government." Although all members of the government are also Members of Parliament (MPs), they are still formally regarded as being the Queen's ministers, as Ministers of the Crown. This harks back to the days when British monarchs had considerably more power than they do today and exercised great influence in determining the make up of each government.
After a General Election, it is conventional for the leader of the largest party in Parliament to go to Buckingham Palace to seek permission from the Queen to form a government. However, this is a a mere formality, a purely ceremonial procedure, and so the Queen is not really in a position to deny such a request. The Prime Minister, once he or she has "kissed hands" with the Queen (figuratively, not literally speaking), can then begin to carry out their duties as head of the executive, the most powerful position in British politics, equivalent to that of President of the United States.
A similar role can be seen in relation to legislation. Before each Act of Parliament becomes law, it must receive what's called the Royal Assent. Once again, this is nothing more than a formal procedure. The Royal Assent can be given inside or outside Parliament. If it takes place inside Parliament, then an official called the Clerk of the Parliaments reads out a formal written document, which says (depending on what kind of bill is being passed) "La Reyne remercie ses bons sujets, accepte leur benevolence, et ainsi le veult," which is Norman French for "The Queen thanks her good subjects, accepts their bounty, and wills it so." For other bills, the form of words used is simply "La Reyne la veult," or "The Queen wills it."
The monarchy in England has been largely a ceremonial post for quite some time now, meaning the Queen of England, Elizabeth II, has no real political power, though she still wields a great deal of influence over the political scene in Britain and the royal family remains immensely popular.
The Prime Minister is much like the President in the United States, in that he/she is the Exective of the country, and passing a law requires his/her consent in most cases, and the PM is also the commander of the British military.