Like indicates a desire to do something; prefer indicates a desire to do something over something else.
I would like to discuss the matter in private.
means that I want to discuss the matter in private, but:
I would prefer to discuss the matter in private.
might mean that I would rather discuss the matter privately rather than publicly.
In short, if you and I needed to discuss something, and I wanted to do so privately, I might say to you:
I would like to discuss this matter in private.
However, for the sake of discussion, let's assume I only said,
I would like to discuss this matter with you.
and we were within earshot of other people. You might respond with,
Okay, let's talk about it.
in which case I might reply:
I would prefer to discuss the matter with you in private.
This nuance can be found in the dictionary. NOAD defines the verb like as:
wish for; want
while prefer is defined as:
like (one thing or person) better than another or others (emphasis added)