Via The Economist (circa 2015):
Those who would like to scrap a popular monarchy need to be able to show that there is a significant demand for a change (which there is not) or that the institution does significant harm, which is just as hard to do. It is accused of being expensive, but offset against the few tens of millions of cost the fact that Britain’s royal heritage is a big part of its tourist appeal, not to mention the unquantifiable but surely substantial brand-management efforts that the Queen in effect performs on overseas trips. An alternative, elected head of state would not be cost-free either.
The monarchy is accused of entrenching elitism and the class system, but it is a fantasy to imagine that those things would vanish in a republic; they certainly have not in America, while the monarchies of Denmark, Sweden and Norway are among the most meritocratic and egalitarian in the world. It is accused of damaging democracy because (on paper) the Queen retains vast constitutional powers. But this ignores the fact that there is not the remotest chance that she or her successors would actually use them; if ever she or they did, then Britain could and indeed should consider becoming a republic.
On the other hand, it is just as plausible to assert that there are benefits to a monarchy, on top of the (hard to quantify) economic ones. At a time when most government institutions everywhere are unpopular and even hated, any part of the state which people still actually like is a rare plus, something not to be discarded lightly. And what would replace the monarch? An elected and therefore political head of state is sure to upset at least one large section of the electorate a lot more than an uncontroversial one who is above politics.
Admittedly, the value of continuity and tradition, and of a focus for Britain’s quiet brand of patriotism are difficult to assess. The reality is that the monarchy does not do much harm and does not do much good; but it is accepted and liked by most Britons. Getting rid of it simply isn’t worth the fuss.