@vessalius
How much power the queen or the crown prince has and can do in the name of the nation varies a lot from nation to nation and scenario to scenario.
Both, can be representatives at some important event, if the king is unavailable.
Both, can effectively assume the role of a minister, but if they're not capable, they tend to have a deputy or vice minister who is (if the deputy or vice isn't capable either, well, then they're screweed)
Both can otherwise advise the king / emperor, even if not part of the council.
There's not really any guaranteed way of saying Crown prince/ Queen is more powerful / mightier / more influential than Queen / Crown Prince, which is true for all monarchies, and if there was, it wouldn't necessarily be beneficial.
Yes, most of our known monarchies are patriarchal, due to one reason or other. Some succeeded better than others.