How to use could and would in a sentence

How to use could and would in a sentence

What is the proper use of “would” or “could” in sentence structure? For example, would you please close the door? Or, could you please close the door? 

I don’t see much difference. But I would suppose that “would” is more polite, because it expresses the idea of probability, and of willingness, and of the desire that something be done, whereas “could” is more in the realm of ability (yes I can). 

And according to the American Heritage dictionary, “would” is used to make a polite request. But then again, a similar thing is said about “could”: “Used to indicate tentativeness or politeness. I could be wrong. Could you come over here?”

Now, as far as I’m concerned, it becomes then a matter of context and tone. Look at the difference between these two sentences, for example: 

Would you do me a favor?

Would you please just shut up.

And even “will” could be used interchangeably with “would” or “could.” 

I guess perhaps “could” and “would” are just both vying for what is a sort of awkward subjunctive mood, trying to put a command into the mode of the hypothetical, to increase the possibilities of expression—toward either politeness or irony (e.g., changing the tenor of “Please close the door”).