With respect... What sort of answers do you get that encourage you to hire the interviewee? What do you actually learn from the answers?
The reason I've always disliked it is that it's just begging for a silly response. Sure, I have weaknesses; but you are asking someone to brag about the things they do poorly at the same time they want to make the best impression. That does not encourage or reward honestly.
It also feels like a "cheater" question, and makes me feel the way I do when a sports news person asks a team member, "How big is this victory for you?" What the heck does she think he is going to say: "I really don't care that we won"?