Two difference between a constitutional monarchy and a paper constitution.
1. A paper constitution hopefully embodies the wisest ideas at a particular moment in time, but will tend to get outdated, especially if it is read in orginal fashion as if it is a tablet of stone. A constitutional monarchy invests in living people and thus depends upon the monarch and his/her family to behave responsibly and represent the wisest contmporary ideas. But if they are incapable of doing so, you have a problem. Strengths and weaknesses in both instances.
2. Both are vulnerable to irreverence. But people in the US don't have to contend with malevolent people like Harry and Meghan whose selfish aims come at the expense of our monarchy. Hence, we vehemently criticise them back. It reduces their ability to do damage.