The difference between would you and could you

would you = do you mind, are you agreeable to, is it OK with you to
could you = are you able to
eg if you could climb that tree would you rescue my cat = If you are able to climb the tree is it OK with you to climb that tree and save my cat.
eg I would if I could = I am content to climb the tree if I were able to


Rather a large question, , but here are some of the uses:

Would

Polite request: Would you close the window, please?
Conditional II and III: If I saw you, I would greet you. If I had seen you, I would have greeted you.
Past habit: When I was young, I would play outside all day.

Could

Polite request: Could you close the window, please?
Polite offer: Could I get you something to drink?
Conditional II and III: If you asked, I could get you a drink. If you had asked, I could have gotten you a drink.

Past ability: When I was young, I could party all night.

Other members will likely come up with other uses of these popular modals.