Nope. Power comes from the Queen, unlike in the USA were power comes up from the people.
In the UK, Canada, Australia, etc. (each with an independent Crown, but all held be the same individual), the Queen "reigns" not "rules" with the advice of Parliament. The PM has no actual power at all: simply one vote in the House of Commons. He is the leader of his party, which is the biggest one in the House (usually; the Queen could choose another as PM, but seldom does), so he can advise his MP's what to do, although he cannot compel them. Granted, the PMO does a lot, but the actual power still resides with the Crown, who will respect the wishes of the people as represented in Parliament, but who still holds the residual power.
Bills etc., must go through both Houses, and then to the Queen for Royal Assent. If she doesn't sign, a bill does not become law. In this, she is like your President. However, unlike your President, she appoints Judges, military officers, and members of the House of Lords (in the UK), the Senate in Canada, etc. This is her direct power, although she will do it on the advice of officials, often the Prime Minister. However, officially, it is her power that makes all this happen.
Other Royals have no power whatsoever. They may have an influence in society, but they have no actual political power.
As for money, she does get some from the taxes raised, but, most of her money comes from her investments, and business. She raises (and sells) racehorses for example, amongst other things. In fact, she pays more from her private monies to run the Monarchy than she gets from taxes etc. As part of his training, Charles received nothing from taxpayers, and had to survive on his income from the Duchy of Cornwall, which is mostly farms rented out to people. Other Royals work as well, in the military, or other places.
Being a monarchy is cheap though. 2009 was the last survey of it in Canada, and it cost $1.53 per person, per year to be a monarchy. The UK is similar; about 84 pence. To put this in perspective, the inauguration alone of Obama cost more than the entire cost of the monarchy in the UK that same year.