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It's not possible to explain it in analogy to sine waves, because instruments are not sine wave generators. If instruments were capable of producing perfect sine waves at consistent phases, frequencies, and amplitudes then yes, they would all sound the same.
Comparing an electric guitar to a piano:
1. Plucking a string causes it to vibrate differently vs. striking it. For a given pitch, this generates differently shaped "main waves", which itself gives the sound different harmonics. Compare the sound of a piano to that of a harpsichord. The harpsichord has more harmonics.
2. The guitar doesn't have a resonance chamber, nor do its strings vibrate the air directly. Instead it uses electromagnets to measure the distance of a piece of metal several thousand times per second. Because the strings vibrate perpendicularly to the pickups, this adds a non-linear distortion to the wave, which adds harmonics.
3. Certain maneuvers are possible in a guitar but impossible in a piano. For example, a piano is not capable of doing a proper glissando, because it can only produce notes at discrete ranges. A guitar is also able to do tremolo, which I think it's impossible with a piano.
4. Finally, different mechanisms of producing sound cause the attack, decay, sustain, and release phases to have different lengths, which also affects the "feel" of the sound.
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