Music making should be a creative act, not just when composing or improvising, but also when interpreting the music of others. Even playing the notes on the page is more than just playing the notes on the page.
Notes are, after all, very tiny packets of musical information. They are important, but a composer doesn't just put them there in isolation, any more than a novelist puts letters on a page without forming them into words and phrases and sentences and paragraphs. And those have meanings often far beyond their immediate sense.
A couple of months ago I mentioned a performance that I had when I was barely recovering from the flu and had had no time to practice. One way I was able to virtually sightread in front of an audience was that I was constantly thinking in larger patterns, rather than having to think about every note I was playing.
Another reason was that I was practical enough to call in Mozart for my little emergency. Music from the classical period (c.1750-1810) tends to have fewer things going on at the same time, and to have these fall into more predictable patterns. There are, for example, more scale passages in classical music than in Baroque or Romantic era music.
Let me give you an example.
Suppose there is a passage like the following:
What it is important to recognize is that the initial downward sweep consists of part of a D major scale. The bottom note is simple the lower chromatic neighbor of the A at the bottom. Then, having reversed course, we continue playing a D major scale until we arrive at the F natural, at which point the pattern becomes a chromatic scale. Now, if you are able to see all that right away, in your mind the passage looks like this:
Now there are only a few things to notice--the remaining notes, which are all important because they introduce changes in direction or changes in pattern. The rest, all prefabricated standard musical patterns, should be on autopilot.
It is bit like making a zip file on a computer, in which vast amounts of data are condensed into much smaller files that can be stored and moved more easily by getting rid of all the unnecessary zeroes--blank space.
Or, you might think of it like being a conductor. The conductor doesn't play every note of the oboe solo--he or she merely cues the oboist about when to come in and how to conceive the passage, giving tempo changes and interpretive clues when necessary, and then leaving the player to do the rest, which is important because now it is time to direct attention to the cello section, and so forth.
In other words, you are the conductor, and your fingers are the orchestra. They shouldn't need you for everything, and conversely, you should know what to tell them to get them started.
Many people seem to want to avoid this way of thinking about the music they play, preferring simply to see the notes and to play the notes, without thinking in larger units, and, indeed, other skills have to be cultivated in order to be fluent in recognizing gestures rather than blindly playing an Eb because Beethoven said so. Creative skills like writing and improvising one's own music have enormous benefit here, so does a study of music theory, although I'm afraid that as it is usually taught theory tends to stop well short of any cross-fertilizing benefits. Perhaps it is because people enter music schools so deficient in theory, but my experience suggests that students learn just enough not to be able to see why they learned it. It would be like getting just far enough to know what nouns and verbs are and then turning you loose to be a writer.
Being able to write music, improvise, analyze, and to think in music isn't just a benefit to the people who specifically do those things, it is critical for the learning and performing of written scores as well.