acoustics - Sound difference between musical instruments

I know that the difference between two musical notes is given by the sound frequency, and the difference in volume is given by the amplitude.

What I am wondering is why does the same note sound different on different musical instruments? What in the wave makes the difference between the sound of a harmonica and the sound of a violin singing the same note?

Answer

The different tonality of a note in different instruments stems from the different mixes of amplitudes in the harmonic frequencies that the instrument provides.

To be more concrete (and keeping to a slightly simplified view), you play the A note (440 Hz) and then you have the harmonic frequencies 880, 1320, 1760, ... ($440n$ where $n$ is the number of the harmonic). Each of the frequencies will have an amplitude or "volume" contributing to the sound produced by an instrument. Thus a particular set of amplitudes gives the instrument its tonality.

This is highly related to the concept of Fourier analysis used in many areas of physics.