I guess the easy way to start is with drugs. Drugs alter normal body functions, like beer will dull the senses, sugar will make you hyper and paracetamol will reduce swelling. Some drugs like alcohol and marijuana do not heal the body and are simply abusive items.
On the other hand, medicines by definition, help the body heal or reduce symptoms. Some medicines may also be a drug, like paracetamol. However, alcohol and marijuana are not medicines because they do not heal.
With that said we can see there is a difference; medicines are intended to heal, like an antibiotic or an antiseptic for the cut. (The antiseptic is a drug because it changes body function, and the antiseptic is not because it only killed bacteria on the skin) whilst drugs may be for healing, some are not for healing.
Some times there is a fine line between drugs verses medicine. For example, looking at the meaning of "medicinal", one might like to have a nip of brandy before sleeping for medicinal purposes. That makes alcohol a medicine, and a drug.