"Music notation" is a contradiction in terms. Music is heard, while notation is seen. Notating music is thus a translation from one sensory modality to another.
Yet in any translation, some information is bound to become lost.
"Written music" is thus an oxymoron. Music is sound, not black notes on a white page. We classical musicians tend to be trained with the implicit ideology of textual literalism—some even going so far as to say that the sloppier Beethoven's handwriting, the more impassioned the music—over music as sound, which is by its very nature immaterial. Yet were it not for written music, we would have neither a preserved musical tradition nor many of its largest and greatest works. After all, a novel isn't composed solely in the author's mind, nor is an entire symphony in its every detail.
Yet sound defies writing down. In practice, very little of it can be written, and what can has been condensed, for all its complexities, into as simple a music notational system as remains practical, developed, with some discontinuity, over the course of centuries. Ways of writing music developed independently in various times and places, from Babylon in 2000 BC to Greece some one and a half millennia later. But music notation defied Western theorists and musicians until about 850 AD.
The dialectic between music as sound (or music "qua" music in pretentious philosopher-speak) and the ability to write it down has only deepened over the evolution of the Western musical tradition, as music and notation have become ever more intertwined. But that's by no means the whole story. In a way, music notation may be thought of as a necessary evil given its many limitations. Western musical notation, as complex as it may seem, is perhaps the simplest way to write down sophisticated musical ideas with a reasonable amount of information that gives us what's most essential yet leaves the rest to our musical imagination.
As a matter of fact, the marvel of modern technology makes it quite possible to notate sound with great accuracy. Sound can now be written down to a level of precision that far exceeds humans' capacity to hear it:
The obvious problem is that such a notational system is entirely useless—there's virtually no way to make sense of this visual information, at least not if our aim is to play certain notes in certain combinations at certain times.
Thus, there's a marked difference between writing sound and writing music. This leaves us with a bit of a philosophical problem, and an ironic one at that. Music is sound, not written notes... yet now that we have the ability to visually represent sound in vivid detail, it's totally useless?
Centuries-old Western musical notation, with far less information, turns out to be quite good after all at representing the vast majority of music in our culture. And it even helped composers write it, which means that more advanced written music—say, a symphony or opera—is more than a mere aide-mémoire. Like a self-replicating genome, music just took both forks in the ontological road, and we can't put our finger on "where" it is: Is Beethoven's Fifth the original manuscript? Is it the first performance? Is it the ideal performance the composer imagined in his mind's ear? Is it some metaphysical amalgamation of its own performance history?
While the essence of music is no doubt sound (a matter of course that is all too easily forgotten by text fetishists attempting to prescribe meaning to every one of Mozart's ink spills), at least some of what music "is" is written notation, just as a literary work "is" a written text.
The goal of music notation is not to put waveforms and decibels onto paper in all their precision, but rather to give performers a set of instructions for what to play, and when and how to play it. Simplified notation is a matter of necessity, and Western notation more or less effectively strikes that balance.