The usual story I've heard describing the difference between a 440 Hz note played by a flute and a trumpet is that the overtones are different. That is, if you play a note at 440 Hz, there will also be Fourier components at 880 Hz, 1320 Hz, etc. The difference in the relative amplitudes of those higher harmonics constitutes the difference between instruments.
However, if you mess around with a Fourier applet (like this one, the first from a Google search), I think you'll quickly become convinced that you can't make a sound like a trumpet, flute, or pretty much anything other than an electronic box with it.
What's the essential difference between the idealized sound source and the real sound source?