I would say that the difference is not that much a matter of formality, but a matter of representing one's attitude to the person in question. An exception is "Rauschgiftsüchtiger". "Rauschgift" is a less modern term and sounds stilted. "Drogen" is the word you will find in common use. Note that, in difference to English, "Drogen" always has the connotation of either "illegal drugs" (most common usage) or "psychoactive substances" (you could argue that caffeine is a "Droge" in this sense, but you will need quite an explanation around it to get people to agree to this usage). It is never simply a "prescription drug" without abuse potential in the sense that in English, a statin is a drug to lower cholesterol.
Out of the three "Drogen-" terms, I would say that, from negative to neutral, they are "Drogensüchtiger", "Drogenabhängiger" and "Drogengebraucher". "Sucht" and "Abhängigkeit" both translate as addiction, but "Sucht" has moralistic roots, with the connotation of "not being able to resist temptation". "Abhängigkeit" is the term more likely to be connected with today's prevalent scientific view of addiction as a medical problem.
"Drogengebraucher" is not something I have encountered before, and seems to be a direct translation of "drug user". Although rare, it might be a better term in many contexts, because it does not make the assumption that the user is addicted. Scholars and activists have long tried to focus more attention on the actual nuances of drug-use-related problems, and to combat the ignorant attitude that all illegal substances are somehow the same, with equal mechanisms of action and outcomes. The choice of weird words like "Drogengebraucher" is thus a political decision, rather than a linguistic one.