Medicine can be a drug, but it can also be a field of study. A cure could be a medicine, but it can be something else as well. He found the cure for cancer and it's not medicine, but nutrition. Modern medicine is based on a reductionist view that states that nutrients isolated in supplement form can replace nutrients in whole fruits and vegetables.

Cure is more of a term for something that can solve a problem, which can be medically related, but doesn't have to be. A cure could the medicine itself, but could also be something else. Here are some examples: A patient is exhibiting signs of an infection but no causative organism can be identified antibiotics and antivirals have been ineffective, and, therefore, did not cure the patient. Strangely, a change of environment healed the symptoms and so it appears that this change was the cure. The cure to a bacterial infection is typically antibiotics. A cure can be used also in the context of a bad experience. For example, the cure to someone having a bad day could be going to a comedy club. Hope this helps