It matters because a hereditary head of state position (actually, any governmental/establishmentarian position) is a terrible symbol in what should be an egalitarian society, no matter if you have political power or not. I don't know how to put this more succinctly.
Why should society be egalitarian? Nature isn't.
Why shouldn't society recognise the heirs of great men specially?
I think that the ideal government has voices for the great mass of the people; for the magnates; and for a monarch. The U.K. system — until Tony Blair destroyed it — was good for exactly this, although the removal of the Lords' ability to block legislation, and the addition of life peers were recent mistakes.
It's a bit more complicated than that, to be fair. A big part of de Gaulle's motivation for the semi-presiential system of the Fifth Republic was due to the instability of the parliamentary system of the Third and Fourth Republics. Whatever else you might say about de Gaulle, the Fifth Republic has been remarkably stable and orderly compared to the regimes that preceded it after the revolution.
The situation in Japan is pretty stable. Why mess with success? And if the emperor has no power, why does it matter if the position is hereditary?