There has been a slow and steady trend in music by composers to be very specific as to their intentions. Whether this is an all together good thing or not depends on the music in question and of course the listener's taste. But the music of history could be very sparse as to performing directions. Even the most basic tempo directions can be very sparse in the music of Bach. And here is one of the mysteries of Mozart's Fantasia in D Minor; It has very little performing directions outside of tempo indications, and the last ten bars are missing. Mozart evidently never got around to writing out the ending of the work or to notate more detailed dynamics or phrasing. Scholars believe that someone else besides Mozart wrote the last few bars of the work.
The piece has three unbarred cadenzas, numerous fermatas, and changes tempo often. The name Fantasia does mean a certain amount of freedom in performance, and with the lack of direction in the piece it assures a variety of performances will happen. And they have. But to the player that is also a scholar, there are indications as to a proper performance by the time period it was written in, the composer who wrote it, and the traditions of the time.
The circumstances that have made freedom of expression so prevalent for this piece have also added to the degree of difficulty of it. If the performer doesn't have the ability to blend the sections into a whole, the seams can be heard and it becomes a string of loosely connected musical ideas that no matter how attractive some of them may be by themselves, the overall piece will suffer from sectionalization. The notes themselves are not difficult. Bringing them together and making music with them is. But that can be said for many of Mozart's works. But this particular piece is somewhat of an enigma, and remains an interpretive challenge for any pianist who chooses to tackle it.
For a more in depth analysis of the piece, I recommend the essay: W. A. Mozart: Fantasia in D minor for Piano - Paradoxes of Style and Interpretation or Fantasies about the Fantasia;by Sophia Gorlin. The essay can be found at her website.
The pianist in the following video is one of the great ones of the 20th century, Claudio Arrau (1903 - 1991), a Chilean pianist. He was a child prodigy and could play Liszt's Transendental Etudes when he was eleven years old. He was known for his exhaustive repertoire. He played all of Bach's music for keyboard in 12 consecutive recitals, performed the entire cycle of the 32 Beethoven piano sonatas many times in public. His repertoire reached from the Baroque to the 20th century. He was known for his deep, rich tone and very introspective playing.