It varies from country to country, both from a strictly legal standpoint as well as a tradition and practice standpoint.
On the face of it, the Prime Minister of the United Kingdom has MORE power over the government than the President of the United States. The PM is Head of the Government, of both the executive branch AND the legislature. The limits on the PM's powers though are of a different nature than the limits on the President's. The PM, technically, is subject to being booted from office at any time. The PM DOES serve at the pleasure of the monarch, although monarchs in constitutional democracies (or their close cousins that function nearly the same without an actual constitution) tend to be VERY reluctant to go out on a limb with their "displeasure" in order to kick members of the government to the curb.
The President has ZERO legal control over what happens in Congress. How much actual influence depends entirely on the political situation If a Prime Minister is opposed by the majority of Parliament, he (or she) will lose the job. A President can have both houses of Congress in the hands of the opposition, yet he will remain President until the next election. In contrast, the power that the PM has over Parliament is as great or greater than that of the Speaker of the House or Senate Majority Leader. The PM selects his Cabinet, and there is no "advice and consent" from Parliament. The PM in the UK can "dissolve the government" and force new elections. The PM himself IS a member of Parliament, with full rights and powers of an MP. Even if he loses the PM, he can remain in Parliament. In the UK, the Prime Minister has no veto in the sense as we understand it. If something is contrary to the PM's legislative agenda, it never sees the light of day, thus, no need to veto.
One of the key fundamental differences is that there is no separation of powers. In countries with unicameral parliamentary systems (Sweden is one), the Prime Minister has FAR more governmental power than ANY individual in the US government. In the UK, there is at least a vestige of bicameralism remaining, as the House of Lords has a bit of actual legal power still.
As AM says, there's a huge amount of complexity out there, not just in the American system, but in others as well. As a general rule though, the American Presidency is actually one of the weaker heads of government. It simply seems like the Prez is so powerful because a) makes for a better media story,

magnified by the power of the US itself.
The great genius of the American system is the separation of powers, a separation that cuts across multiple dimensions. That is, of course, also one of the greatest sources of annoyance. Gridlock as we experience it in the US rarely happens in other countries. They are more likely to be plagued by paralysis or revolving door governments. (Note:
Bureaucratic gridlock is a different story, and plagues EVERY country.)
Oh, and the British PM can start a war just as easily as an American President. (The United Kingdom, like the United States, has enjoyed a period of unbroken peace since they last declared war during WW2. Contrary to the fiction you'll see peddled about, there were no British troops engaged in the Korean Conflict. No British troops were involved in any fighting during the assorted African and Asian wars of independence. The British involvement in the Suez Crisis was actually a Mossad black-op, run by some shadowy figure named "Zohan." The entire dustup over the Falkland Islands was simply a staged movie collaboration involving Kubrick, Spielberg and Lucas. All of the "British" troops who fought in the First Gulf War, and subsequently in Iraq and Afghanistan were actually Blackhawk contractors hired by the CIA. Bosnia? Kosovo? Nope, no True Englishmen were involved. Libya? Syria? ) By definition, the PM has the backing of the majority of Parliament. The key difference isn't their respective AUTHORITY, it's their nation's CAPABILITY. The UK simply lacks the ability to project and exercise power to the extent that the US can.