A great way to hear overtones in action is to look not at acoustic guitars, but at ride cymbals. There is a fundamental "ping" when a stick strikes the cymbal, followed by a "wash" of overtones that paint the overall sonic landscape of the cymbal. Different styles of music call for more or less fundamental tone vs overtone in a ride cymbal.
Going back to acoustic guitars, those ringing overtones can really fill space beautifully, but the can also get in the way if the music doesn't call for it. For me, I enjoy the sound of rosewood a *little better, but mahogany works a *little better with my vocals (the overtones fight my ears and pitch control... But just a *little)
