@Dawson,

Alex and Alexandre gave perfect explanations of what these different words mean. Whenever I hear Droge, I think of just street drugs, like Heroin or something. When I hear Medikament I think of prescription drugs ordinarily, although an Apotheker (in) can dispense some things without a prescription that you aren’t going to find in a Drogerie. Those are the things we would call “over-the-counter” in California. I have something I got from an Apotheke last July that says right on it “Apothekenpflichtig!” I don`t know about the legalities of it, but it appears the things in the US you pick up and buy over-the-counter, can only be obtained at an Apotheke in Germany, and it may require the pharmacist actually agrees to give it to you.  A Drogerie is for like hair products, toothpaste, shampoos. Non-medical stuff. They sometimes are pretty big stores too, while all of the Apotheken I`ve been in look really tiny by comparison. You walk in, and there’s very little strolling around to do. 

If you get to Germany you will very quickly learn the differences in how these terms are used, and how drugs are dispensed. In California, you can walk into just about any supermarket and find a pharmacy (Apotheke). So, you just drop off your prescription in California, then go do the rest of your grocery shopping, all in the same place. Not so in Germany, as far as I`ve ever seen. But much of the same stuff you can get at a Drogerie you can get at Edeke or Lidl, or Rewe or some supermarkets like those. They`ve not got anything like a CVS, that sort of mixed Drogerie and Apotheke all in one, where you can drop off your prescription, then stroll around and pick up some mouthwash and shampoo while you wait for your prescription. 

I think Germany is MUCH more rigid about prescription drugs than the US. There are medicines here you can buy at a supermarket that require a prescription over there.

To give you a rough idea

Droge drug (
Medikament medication
Arzneimittel medicine
Rauschgift drug
Betäubungsmittel narcotic, anesthetic

This dictionary might be of some help: <a href="https://dict.leo.org/german-english/droge">https://dict.leo.org/german-english/droge</a>;

drug store means Drogerie in German.