Suppose you want to ask someone to do something, say to tell you the way to a station. Which is more polite, "Would you tell me the way to the station?" or "Could you tell me the way to the station?"?
When you say:
Would you tell me the way to the station?
you are asking a favor of someone.
When you say:
Could you tell me the way to the station?
literally you are asking if the person knows the way to the station and can tell you. As a question it is implied that you are hoping for the answer, but permission or whether the person wishes to answer is not addressed.
As such, using would is more polite.
Tentative volition: Would you tell me the way to the station?
Tentative possibility: Could you tell me the way to the station?
If you think of a case to decline the request, to answer a "Would you...?" question, the literal answer to that question is "No, I would not." This sounds like you have no will to help him/her, and so it is a bit awkward. As such, saying "No" to a "Would you...?" question is rather difficult at least for a kind person. If you think about saying, "Sorry, I could, but I wouldn't," it almost sounds like an expression of hostility. In a sense, you are not giving a choice when you ask a "Would you...?" question.
In contrast, declining a "Could you...?" question is relatively easier. You can blame the weather or something that you cannot control and say, "No, I can't." or "Sorry, I can't." You could even say, "No, I really would, but I can't, sorry." You don't have to feel that awkward. Because you are making it easy for the other to say "No" when you are asking "Could you...?" or "Could you possibly ....?", these questions are more polite than a "Would you...?" question.
Maybe it would up the politeness level if you said something like "Could I trouble you to...?" or "So sorry to bother you but could I ask you/could you tell me/would you be able to...?" or something along those lines, acknowledging that it is possible they are able but you don't want to assume they are willing.