I think there is an important but at times subtle difference between
overtones and harmonics. On stringed instruments and organ pipes these
two ideas often overlap.
Harmonics are integer multiples of the fundamental, but for a given
system these may not be present in the sound produced.
Overtones deal with the sound actually produced. An example, given by
Wikipedia, is a drum head in which case the first overtone is 2.4 times
the fundamental. Another example is brass horns. Their overtones are
shifted a bit higher than the harmonics of the fundamental, because the
shorter wavelength sound decouples from the flare of the horn earlier in
the flare. (The pressure node of sound wave at the open end organ pipe,
is actually just beyond the end of the pipe. When the wavelength of the
sound wave becomes comparable with the diameter of the pipe or the part
of the flare things get complicated.) This shift of the overtones from
the harmonics produces the characteristic "bright" sound of a brass horn.