I have this argument with my grandma constantly. I don't see the Royal family server any useful purpose, other than the occasional tabloid frenzy when someone dresses up as a nazi or they sneak a reporter into the queens shower.
I suppose the idea is that they are an example of all things British, dignified, classy. If you really want an example of Britishness these days Johnny Vegas should be our king.
I don't object to the idea of a non political ambassador to the rest of the world, but I don't think the Royal family are the best choice. The fact that the monarch is born into the role is a big problem... you never know who you'll get and you can't get rid of them once they're there. Maybe a stable celebrity ould be voted into the office occasionally. I don't mean the borderline neurotics you get on the front page of the tabloids everyday, but there are more suitable celebrities out there who might actually be pretty damn good at it. Rowan Atkinson, Bob Geldof to name a couple and I'm sure there's more.
In anycase the monarchy in its current form should go in my opinion.