Both of these names refer to higher ranks than the bishop in the Roman Catholic Church, but archbishop is more obvious to most people and thus became more popular.
Christian Freeling, the inventor of Grand Chess, took a similar approach to Capablanca, naming the piece the cardinal.
By analogy with the queen, which is a rook+bishop compound, it was decided that the three basic combinations of the three simple chess pieces (rook, knight, and bishop) should all be named after female royalty.
A mathematical approach used to determine relative piece value in Musketeer Chess estimated the value of the princess as 770 centipawns on an 8x8 board.
However, all three of his alternate armies for that game are stronger than the standard FIDE army which they were supposed to equal, reflecting the general tendency for players to undervalue pieces that they are unfamiliar with; Larry Kaufman commented that this is particularly true for the princess.