Music for the piano has a unique quality. Along with the organ, it is one of very few instruments that can present music that would otherwise require a whole ensemble of players. The piano can handle two part inventions, three part fugues, four part fugues, even five and six part fugues! Composers have transcribed the music of string quartets and full orchestras for the piano.
Of course it is not always physically possible, even with ten fingers (and possibly two feet for the organ) to play every independent musical line that would be played by a dozen instrumental groups in an orchestra…but the piano can convincingly create the effect of a large ensemble in a way that melodic instruments cannot even begin to dream of doing on their own.
So what does that mean about how you should think about piano music and playing the piano? Obviously, if you want to play several independent musical ideas at the same time, you have to have full control of your fingers. But even more importantly, you have to have the capacity to recognize and follow all of these musical layers as you play. Your job is rather like that of the conductor of a symphony orchestra in that respect. But it also has the added dimension that you are the one playing as well as directing!
It can be so tempting for pianists to revel in all the notes they get to play and pound them out as fast and furious as possible. It is easy to impress people this way, but if you want to make your piano playing meaningful and beautiful, think like a conductor.
Imagine what instruments are playing each of the musical textures that you find on the page. Is that melody an airy flute solo or a throaty saxophone wail? Is there a viola solo hidden in the inner voices, waiting to burst out for its moment in the limelight? Exactly how loud should those cellos be playing the bass line? And like a conductor, you must coordinate your musicians and get them to play precisely together. And help them find the architecture in the music. Then, when you have done that and are aware of all the layers that make up the music, you will have the freedom to be spontaneous and expressive as you play.