The royals bring close to a billion a year from tourism and land rights into the British economy. Any kind of tax we pay is multiplied many times and put back. And thats ignoring the best thing about them, The queen is the best ambassador that any country will ever have (well maybe the Dalai Lama)
The Queen has no real power, therefor she can't be blamed for anything the Prime Minister does. Sending the queen on a Good will mission to other country does nothing but good usually, Sending Someone like Cameron or Obama to a country can have negative side affects, regardless of whether or not they did it, they only see the war in Iraq or Afghanistan, Protests and the like. With the queen they see a harmless but perceived to be powerful old woman representing some ideal view of the UK. It does nothing but broker good will among the country shes visiting (regardless of some apathy) and helps massively with British business deals and the like.
I like them, some of them are Grade A Hunts but oh well, for there Flaws Harry and William seem to openly care about when they do, Harry enough to quite the press so he could fight on the front lines of a War.
For all the republican feelings and random press silliness (and the DoE's inability to not be randomly stupid as all hell) they do far, FAR more good for the UK than anything else.