I do think it's important to think about both the overtone series, and perhaps even more importantly finding a good ringtone. I'm glad I've finally been given the knowledge of where to look for them. Thank you, thank you!
Seriously though, it can be very useful to think about harmonics. As
(@PaperComposer SORRY I CAN'T GET RID OF THIS TAG)
@Polaris said, choosing any instrument involves the overtones. When writing for combinations such as the wind quintet, voicing chords can be greatly helped by knowing which harmonics emphasise each other. You can even create the effect of there being more notes than there are.