It's probably true that 99.9% of music listeners don't know, don't care, and can't tell the difference. But a few of us care a lot, and it's sometimes very important. I've been recently obsessed with barbershop quartet music, where the tuning of chords is absolutely pivotal. The alignment of overtones from justly tuned chords is, if you ask me, the entire point of the genre. The "ringing" produced from the aligned overtones is, I would guess, noticeable to a large portion of people with rudimentary music education.
https://en.m.wikipedia.org/wiki/Barbershop_music#Ringing_cho...
Two examples of fairly audible ringing (probably start a little before my timestamps to get a little context):
http://youtu.be/KpBRetTqnqQ at 3:07
http://youtu.be/z9L4QEptVRo at 4:23