When you get into the scientific study of psychoacoustics of how pitch and consonance are perceived, you'll need to consider harmonics. Except for some synthesizers that generate pure sinusoids, all pitched instruments produce sounds that include significant overtones that are an important part of pitch perception. The main theory for consonance is that it is related to the perceived "beating" between the harmonics in the two sounds. So if you get into the quantitative study of this, or are just generally curious about how humans process sounds, then harmonics are a key feature to consider in explaining/clarifying these phenomena.
There are some sonic "magic tricks" that rely on understanding how partials are perceived, and thus can be manipulated. These include the missing fundamental effect, and Shepard tones.
If you consider polyphonic singing, which I consider magic, you probably don't need an intellectual understanding of overtones in order to execute or appreciate it. But if you want and idea of "how does she do that?", then you'd need to consider harmonics.
In terms of piano practice and performance itself, this kind of information is not really required (millions of pianists have been successful without really understanding any of it). If you get into piano tuning, it might be nice to know why the tuning is stretched, but as I understand it, in practice you want to do it by ear anyway.