Depending on the instrument, one (or quite possibly all) of the following complicate things:
- There are a lot of overtones.
- There are sounds other than overtones (percussive sounds and the like), at least with some instruments.
- Everything changes over time in the course of a single note, and quite quickly.
- Successive notes have different overtones, different other sounds, and everything changes in a different way, at different speeds and at different times than it did in prior notes.
Actually, there are some sample-based grand pianos which sound pretty good to my ear, if not exactly like the real thing. At the single-note level, a piano is actually relatively simple compared to, say, a woodwind or violin: the hammer always hits in the same place, and the strings aren't manipulated with anything other than the hammers and the damper (except in the case of some notably "modern" compositions, or possibly Jerry Lee Lewis). The complication in the piano comes from the fact the player is hitting lots of keys at a time and in rapid succession, and the strings aren't independent, isolated devices, but they're all lying in reasonably close proximite in a box, pressed against the same soundboard.