So far I have not seen another possible distinction between overtones and
harmonics.
Harmonics as the name implies are "harmonically" related to the fundamental.
So they would be even multiples of the fundamental. As such harmonics are
often generally pleasing to the ear. This is why several percent harmonic
distortion is often inaudible. It is covered by the harmonics in the music,
and does not sound bad.
The word overtones implies tones above the fundamental, but they do not have
to be related by whole numbers, or even rational numbers to the fundamental.
Some instruments put out a variety of overtones which are not harmonic. The
first instrument which came to mind is the krumhorn which produces a buzzy
sound. It definitely has a pitch, but the overtones may consist of both
"harmonic" frequencies and non harmonic frequencies. Modern instruments
are generally designed to be more harmonic, but older ones often produce a
variety of sounds. Many percussion instruments probably produce non
harmonic overtones, at least in the initial attack. Traditional flutes may
have both the musical tones, and a breathy sound which may not be harmonic.
The musician's word for what you hear is timbre. This encompasses all
perceptible quality of the instruments sound.
I also seem to recall that there may be some instruments which put out
mainly harmonics, but fool the ear into thinking the fundamental is there.
But I would say that mostly people refer to harmonics and overtones as if
they are synonyms.
Compressed lossy music formats generally remove many of the non harmonic
frequencies, as I understand. An audio professional who has a lot of
experience with them claims that he can hear the difference. So even though
non harmonic frequencies do not show up on a scope, they may be readily
heard at even low levels. Surprisingly one of the types of audio which was
the most difficult to compress fairly "transparently" was a male speaking
German! Music seems to be easier.
Supposedly synthesizers can imitate the harmonic structure of instruments
very closely, but they do not sound the same. Could this be due to non
harmonic frequencies being absent?