By definition, distortion is the process which in some way modifies the audio signal. We can say that any process applied to the audio that modifies the waveform (EQ, chorus, phaser, etc) can be called distortion.

Harmonic distortion is a process that modifies the signal in a non-linear way, and consequently, generates new content in the audio signal that previously didn’t exist. This new content generated is usually referred to as “harmonics”,

Harmonics

When an instrument is played, the pitch we hear is the fundamental frequency, and it’s the first harmonic. There are many other harmonics, and these help to shape the timbre of the instrument.
Harmonics are a set of sine waves that form a single one. Any audio source can be represented as a sum of sinusoids waves.