Per the definitions you link to, a quandary is:
a situation in which you are confused about what to do
This could be any sort of situation where the "best" choice is not obvious.
A dilemma is:
a situation in which you have to make a difficult choice
The intent here is that every option available to you is undesirable, but you nevertheless have to choose.
Often, when you have a dilemma, it is also a quandary, because which of the bad choices is "least bad" is not at all obvious and it is difficult to decide what to do. (This is not always the case, though; sometimes you might easily choose a "known to be about X bad" over an "I have no idea how bad it could get" option.)
On the other hand, when you have a quandary, it is entirely possible to be picking from a set of choices that are entirely good in different ways, or that balance good and bad aspects, or that are a relatively meaningless set of choices that you just can't decide between.
Picking socks may constitute a quandary if you simply can't determine which set of socks will best match your outfit, but it is definitely NOT a dilemma, where no matter what socks you pick your life will be difficult and the only question is how it will become difficult.