Most of the European Monarchs are figureheads. They bring in some tourist dollars, although I have a feeling that the Brits are the country that gain the biggest bang for the buck having royals, and they are a nice conncetion to the countries history.
I have a sneaking suspicion that if any of the European Royals exercised what political power would find themselves out of a job as Parliament would vote them out of existance.
So the Queen meets with the PM once a week in GB and is kept up to speed with what is happening but she has no real say in policy or the political agenda. Her speech to Parliament at the opening is written by the PM's staff and she delivers it.
I am not sure what powers the Monarch has in Belgium but I would guess that the Monarch is very careful because all it takes a mistep or two for Parliament to decide that a Monarch is not needed.