Part of a person, is his physical body. That body can get sick in the organs, including the brain leading to mental illness,which can be treated through medicine.
Addiction is in its essence a spiritual malady, a disconnection from the soul, which manifests itself ultimately in the inability to control drug usage, more importantly , bad character i.e. extreme selfishness, dishonesty manipulations, anger,( which develops in choices made through free will, a function of the soul).
We see symptoms of mental illness common in addicts. but a closer look they are also part of the root. An addict is “self will run riot”If everything doesn’t go his way, he’s depressed or angry,and anxious that his dishonesty will be revealed. These problems go away once he is connected to Hashem ,and his character is fixed.
In short the difference, is the bad character synonymous with addiction. Which means it has something to do with the soul. As character develops from free choice, and there is no scientific proof of free choice (In fact scientists argue against free will ).