I just got off the phone with a customer service representative who used the term and totally p****d me off. I was already in a bad mood because I needed to call customer service because of a product problem. Her telling me that "It was no problem" just rubbed salt into my wounds. If it were no problem, I would not have had to call her and we wouldn't have been having the conversation.
It WAS a problem. I wanted her to deal with it, and not dismissively.
So I would recommend that you don't ever use either version of the statement.
I think it is appropriate to say "not a problem" if you are helping me and you want to assure me that the problem is indeed fixable. However, that is a rare situation. If you are representing a company that has caused my problem, saying "no problem" is annoying. Had your company not caused my problem, I would not be here trying to resolve it.
Also, I agree with Lemony that if you are not able to help me solve my problem, saying "not a problem" is maddening. I also had someone from customer service say "not a problem" after each thing I said. I called my insurance company to report a claim after a car accident that caused injuries and left me very shaken. After the young woman's fourth "not a problem," I said "please stop saying that. It definitely is a problem. I just got into an accident that resulted in several injuries -- it is a big problem." She apologized and only said it one time (a forgivable slip) after that.