There's been somewhat of a trend in that direction since Pope John Paul I in 1978 kicked it off by downplaying some of the royal-seeming aspects of the papacy:

There's been somewhat of a trend in that direction since Pope John Paul I in 1978 kicked it off by downplaying some of the royal-seeming aspects of the papacy: he switched to using "I" instead of the royal "we", and opted for a simpler inauguration ceremony instead of being crowned with the papal tiara in the traditional coronation. John Paul II continued some of that trend towards more informality, though Benedict moved a few things back in the other direction (not so far as to bring back the tiara, though).