At first, I didn't recognize any big difference. We learned at school that both expressions are equally polite. But he insisted in his book that if you constantly use "Would you 〜？"in your daily conversation, you might not be able to maintain a good relationship with your spouse or boyfriend.
Here is his explanation: "Could you〜？" originally comes from "Can you 〜？", which is a politer question, but it also implies whether you have an ability or not.
So, "Could you take out a garbage?" has a slight nuance that "You have an ability to take out a garbage and why you don't do that?" Thus, there's a possibility that men might take it as a criticism.
I'd like to ask male native speakers of English?
If your girl friend or wife ask you as follows, do you feel bad?
"Could you bring a beer ?"
"Could you wash the dishes?"
"Could you fix our lunch?"
"Could you drive a car?"
and on and on everyday...
Is his explanation valid?