In America, if it's prescribed by a doctor we usually call it "medicine". If it's advised by your grandmother, we usually call it a "remedy". Gargling with salt water and honey is a remedy for a sore throat. Antiretrovirals are medicine for AIDS.

If you're talking about something that you apply to the body internally or externally to alleviate some symptom or condition, then they mean the same thing. They're equally formal, I'd say. But the word "remedy" in general means something that solves a problem. Medicine in general means the science of the functioning of the body as well as its diseases and conditions. So you can see how those two words have been applied to the special case of something that makes you feel better when you're not well. :D

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