Head of State
A key feature that differentiates the UK from many more modern constitutions is the presence of an unelected Head of State (the monarch). Many people object to this as undemocratic and perpetuating privilege.
Democratic deficit
If the Head of State exercised any significant power, the democratic objection would carry some weight. In most countries where they have an elected Head of State, that role involves significant powers, often through its combination with the leadership of the executive branch of government. However, in the UK, the role of Head of State is largely ceremonial. The last vestiges of authority - the royal prerogative - are exercised with the knowledge that using them against the democratic will would see the monarchy rapidly brought to an end. If the greatest intrusion they make into public issues is to support a couple of contestable scientific theories and to speak up against the modernist architectural establishment, then this is hardly powerful evidence that democracy is undermined by the presence of an unelected Head of State.
Privilege
The strongest argument for abolishing the monarchy, in the opinion of this author, is the opposite of privilege - the awful position in which it places members of the Royal Family, duty without power, foisted on them purely by accident of birth. Yes, they have access to material comforts that most of us could not imagine, but in exchange they are obliged to spend their lives in public, attending charitable events, making small-talk, and pretending that they have no opinion on issues about which the rest of us would express ourselves freely. How many of us would sacrifice our privacy and freedom of movement, career and expression for a life in grand settings? But if they are willing to do so, sympathy is no argument for preventing them from living a life that they are prepared to accept.
Cost
The Royal Family cost taxpayers around £40 million per year. That is around 66p per person, or 0.006% of projected government spending in 2009-10. We would not save £40 million per year if the monarchy were abolished, because someone would have to carry out the official functions of the Head of State, with the associated costs of travel, hospitality and security.
Value
The monarchy does little harm and some good. If we combined the roles of Head of State and Head of Government, either our Prime Minister would have to spend more time on ceremonial duties and less time running the government, or we would be less well-represented at official events. There is the perhaps somewhat overstated benefit to the tourist industry of the persistence of our traditions (though one might expect that people would continue to be interested to visit the sites of the royal estate even if the royals were not still living there, and might enjoy greater access). One should not underestimate the importance to our military personnel of fighting for queen and country. And there is value in differentiating the country (and its formal representation) from the political party that forms the government of the day.
Conclusion
So long as the position of Head of State is effectively ceremonial and the head of the government is democratically elected, countries (including the UK) should not lightly change the arrangements that have evolved from their history. Provided that it does not cost the taxpayer much and does not intervene in democratic politics, the monarchy should be retained as Head of State in the UK.