Originally Posted by Damien

Not just that but it's a great thing to wheel out for visiting Heads of State and other important people. I think we underestimate the appeal of a visit with the Queen at Buckingham Palace to people outside the UK. Trump wanted May to pass on his regards to the Queen when they talked (although given it's Trump it's not entirely clear he knows the Queen is more of a symbolic position). Obama and especially Michelle Obama seemed to be quite fond of their visit. The Queen's visit to Ireland was very successful too.
The Royal Wedding was a big event for international news networks, took front pages across the world.

In purely pragmatic terms they're just good and useful ambassadors for Britain and have a level of status that can't be matched by Prime Ministers. Not to mention the fact they're not political means they can transcend that and represent the country in a way divorced from it's current political representation.