Well depends how you look at it. They are pretty much the same in many cases. There are plenty of "drugs" that aren't addictive and plenty of "medicines" that are. Is caffeine a medicine or a drug in your view? I don't see as much of a distinction as you would appear to.
Again, I'm not saying there is much difference - just that there are languages that use two separate words for those concepts and don't have a (common) one for both.
For example, in Polish you'd classify aspirin as a medicine ("lek", "lekarstwo") and caffeine as a stimulating substance ("używka"). There is another word that is used to refer to harmful addictive substances like cocaine ("narkotyk"). There is no generally used word in Polish that encompasses all three terms.
There is a lot of overlap in these two sets.