In my opinion, the way Mozart should be played on the piano has a lot more in common with Chopin than with Bach, Haydn or Beethoven. The clunky period-instrument Mozart with an almost non-existent sound and a constant staccato has almost nothing to do with what I think he wanted his music to sound like. That kind of playing, while historically correct, is far from the spirit of the works and makes Mozart's piano music sound completely different from his music for other instruments and for voice. Mozart's work for winds, strings, orchestra, and voice are as enlightening in regards to interpretation and sound on the piano as musicological research.
There are three things that I find Mozart has more in common with Chopin than with other composers of his time: (1) a constant singing line, (2) fast passages that are melodic in nature, and (3) a way of developing accompanying harmony in a horizontal way.
Both composers were very particular about a singer's capacity to maintain a legato line. In Mozart's letters to his father and Chopin's letters to his own family, the both make nearly identical remarks about the interpretation of grupetto markings (they should be integrated into the musical line, instead of being mere ornaments--- Chopin was pretty upset after listening to a Bellini performance where the singer kept singing these melodic figures with an accent on the first beat and a diminuendo, instead of having the grupetto lead towards the next note) and both composers also write melodies which carry almost a full harmonic weight by themselves--- most of the time, the melody can stand alone. In the interpretation of both composers' work, there also needs to be a larger than normal distance between the dynamic of the high singing voice and the accompaniment, giving the illusion of space. A common mistake that many pianists make with this music is having the left hand part strangle the melody, especially when the melodic line has long notes.
In both composers, fast passages usually have implied harmonic movements that are quite intricate. In all cases, these passages should be studied slowly and with attention paid to phrasing because, in the best performances of their work, these passages never sound like mere technical exercises. With today's technology we can listen to the recordings of old masters like Rubinstein or Hoffman at 1/4 or 1/8 of the speed without distorting the notes. It is very interesting to do that (with aid of the computer or audio player) because, even when the passage is very fast and complicated, it is still phrased nicely with attention payed to the harmonic and melodic progression in the music. Few pianists today can pass that test, a few are Arcadi Volodos, Murray Perahiah, and Krystian Zimmerman. In the music of Chopin, these fast passages are usually a consequence of the musical line, as seen very often in his nocturnes, and should not be played as something independent from the rest of the music; because these passages are usually studied by themselves, this tends to happen a lot. The same thing happens with Mozart, although the melodic and harmonic lines are usually more simple because of the differences between the classical and the romantic styles.
Alexander Satz used to say that, in Chopin, our most important finger is the thumb of our left hand. The same applies to Mozart. The way both composers develop the harmony in a simple accompaniment usually leads to a secondary melodic line that is played almost entirely with our left thumb. Both composers also tend to use an accompaniment that is much more complicated than the simple arpeggiation of the chords or a simple basso de Alberti favored by their contemporaries.
A perfect example of the similarities that should exist in the interptretation of Chopin and Mozart is in Clara Haskil's recording of Mozart's concerto number 23 coupled with Chopin's second concerto. This recording is a must-have for any pianist. While stylistically, the performances from that period are not considered acceptable these days, I firmly believe that pianists from her generation and before did a much better job of doing justice to the spirit of Mozart's music than most of the playing that we are used to hearing today.