I’d imagine she tells the Prime Minister what she thinks considering she has regular meetings with him.
I hope that one day King William has a strong opinion on policy, they try to strip him of his power, and so he runs for Prime Minister. 
As Bagehot wrote, the British monarch has “the right to be consulted, the right to encourage, and the right to warn.” But when you’ve held the throne as long and as successfully as Elizabeth II, your influence can be quite broad.
To go a little further afield than Europe, the King of Thailand is quite influential even though his actual powers are limited. When he didn’t condemn the military coup a few years ago, it was able to consolidate its hold on power in a way it surely wouldn’t have if he’d condemned it (as King Juan Carlos of Spain did in an analogous situation in 1981) : 1981 Spanish coup attempt - Wikipedia More’s the pity. Thailand also criminalizes criticism of the monarchy and jails people for it from time to time.
As has already been noted, the monarch and some other members of the royal family are granted what amounts to an annual salary, although it may be given a special title depending on the country (it’s “apanage” here in Norway, too). Certain functions, such as security, are usually funded from the national budget rather than the royal family’s own, and those properties that are owned by the state are also maintained at state expense. However, most, possibly all, royal families also have land and other property of their own, which they must pay for the upkeep of - but which they can also derive income from, for instance by renting out the land to local farmers or by having the land farmed themselves.
Thanks for the correction. It still serves as a good example of jointure of disparate powers like the U.S. President’s though, since what I said was true for several hundred years before 2006.
Maggie had a reputation for arrogance and superciliousness towards others. That was what I was referring to, along with some innuendo/gossip about her supposed attitude toward one of the Queen’s sons on an issue of architecture and the Queen genteelly making a superficially-bland statement that was supposedly really a slapdown of the P.M.'s purportedly-disparaging remarks. In any case, I didn’t intend to make a point about Mrs. Thatcher vis-à-vis the Queen but rather to her (real or presumed) egotism – and it was just a sidelight to the main thrust of my post.
There’s generally a government allocation – the appanage mentioned in other posts. the ‘Civil List’ in the U.K. – set aside in the national budget for the support of the Royal Family in constitutional monarchies.
It’s interesting to note the case in the U.K.: The country actually makes a profit on the monarch, in that in 1952 Elizabeth deeded over the revenue from the Crown Lands (but not the Duchy of Lancaster lands) to the Government for the duration of her reign, in exchange for the Civil List allocation, following her father’s and grandfather’s example – and the income from them is somewhat larger than the Civil List. On Elizabeth’s death, Charles will have to decide whether ro follow her example or refue the Civil List and “live off his own” – i.e., from the income from the Crown Lands.