Well, never having actually seen the protocol, I could be off a little. It may be Catholic royal ladies that are essentially the political figureheads of their respective countries. For example, if Prince Albert of Monaco were ever to get married, his wife would be the Princess of Monaco and if she were Catholic, she could wear white when seeing the Pope. At least that is how I understand it. Not being of royal lineage nor likely meeting the Pope any time soon, I don't really need to worry about it. ;-)
As I understand it, the House of Savoy are the claimants to the Italian throne, hence the right to wear white. The Duchess of Luxembourg has the right by special decree.
What do I know? LOL! I found this article with a picture of Princess Grace and Prince Ranier of Monaco in St. Peter's Square after a private audience with the Pope (John XXIII?) and she is wearing black.
I have read that it is only Catholic queens who have the privilege of wearing white. As you pointed out, they don't have to be currently sitting queens and one may obtain a special permission to wear white in the presence of the Pope.