A pain (une douleur) is anything that hurts. A backache is a pain in your back. A toothache is a pain in your tooth. If you have a pain somewhere, you can take 'pain relief' such as aspirin.
A wound (une plaie) is when the skin is opened. If someone shoots you, you have a bullet wound. If someone stabs you, you have a knife wound. You will have a wound after you've had an operation.
'Sore' is usually an adjective. If something is sore, it is mildly painful, and often sensitive, irritated and/or inflamed. For example, if you have a throat infection, your throat is sore. If you have walked a long way, your feet will be sore.
'A sore' as a noun translates as 'une dartre'.
I hope that helps. Please ask if anything's still not clear.