I think the critical difference lies, not in harmony or in finger dexterity at all, but in the internal perception of rhythm. Jazz, both old and new, demands a very highly developed natural sense of rhythm which transcends musical notation to a much greater extent than does classical music. Because of its complete dependence on a sophisticated sense of rhythm in the player, jazz is far harder to approximate in notation than classical. In those areas where fairly accurate transcriptions have been made, for example Waller/Farrell, Waller/Posnak, Morton/Dapogny, Brubeck/Brubeck, a classically trained pianist will certainly play all the notes without experiencing the difficulty of a Liszt study. I hesitate to even go that far actually; a Waller solo like Gladyse, or Morton's Fat Frances feels like a syncopated Mazeppa in places. However, as Thal rightly points out, very few such players understand rhythm at a deep enough intuitive level to meaningfully interpret the music.
For me, jazz and classical are both subsets of the huge landscape of piano music. I enjoy playing it all and listening to it all to the limit of my time and ability.