Harmonics and overtones oftentimes get confused for one another because they share a lot in common and both relate to the frequencies of audio. Let’s try to clarify the subtle difference between harmonics and overtones with this quick comparison.
What Are Harmonics
As I explained in my overview of what are harmonics in music, harmonics refer to the fundamental frequency of a sound and its higher octaves.

So if, for example, we play an E2 note on the piano which rounds down to vibrate at 82Hz, its harmonics would be that E2 (82Hz), the next octave up of E3 (164Hz – 82*2), the next octave up of E4 (328Hz – 164*2), and so on. Note that these are rounded down frequencies (the specific frequency of an E2 is 82.40689).
Virtually every noise you hear has higher harmonics which ring out as part of that note, an octave or more above that fundamental frequency.
What Are Overtones
As I covered in my overview of what are overtones, overtones are frequency resonances at and above one octave above the fundamental frequency.

I mentioned in that overview, but overtones are essentially a catchall term for any frequencies which resonate as part of that sound starting at that octave up above the fundamental.
Difference Between Harmonics and Overtones

In simplified terms, the difference between harmonics and overtones is that the harmonics of a sound relate to the fundamental frequency and its octaves (like in the previous example of playing an E2, the E2 itself, E3, E4, and so on and their respective doubled frequencies), but overtones begin one octave up from the fundamental and encompass other frequencies heard as part of the sound.
Strictly speaking and by definition, every harmonic of a sound doesn’t necessarily resonate whereas all overtones do.
This is just the annoying semantic reminder that a plucked E2/open low E string on the guitar (for example) technically has an E9 as a harmonic some 7 octaves up above its fundamental root, but it won’t be resonating as part of the sound.
Partial semantic differences aside, ultimately, BOTH harmonics and overtones are part of the frequency makeup of your audio.
Check out my tutorials on both saturation and EQ for information on how to massage, adjust, and even add harmonics and overtones in some cases to your audio to solve all kinds of tonal issues in your mix.