tl;dr: An opinion is not falsifiable (e.g. "I like X better than Y"). An answer is (e.g. "X is better than Y").
The term "opinion" is an umbrella that contains two sort of statements:
I think this is really what Robert Cartaino was touching on in Good Subjective, Bad Subjective. An opinion can be grounded in facts, explaining the how and the why, a thoughtful analysis that weigh each factor carefully - or it can be a mere statement of one's tastes.
The best display of the difference I can remember of right now is Roger Ebert's I'm a proud Brainiac blog post, where he does show there are two kinds of opinions:
So let's focus on those who seriously believe "Transformers" is one of the year's best films. Are these people wrong? Yes. They are wrong. I am fond of the story I tell about Gene Siskel. When a so-called film critic defended a questionable review by saying, "after all, it's opinion," Gene told him: "There is a point when a personal opinion shades off into an error of fact. When you say 'The Valachi Papers' is a better film than 'The Godfather,' you are wrong."
Ebert makes a distinction between a good film and enjoying a film. The former requires an analysis of the craft behind the film; the latter is a summary of your own personal experience.
An analysis of the craft behind the film can be wrong. There is certainly room for healthy and reasonable differences of opinions, but past a certain point you're just flat out wrong. There are some films which are simply masterfully executed and other films which certainly are amateurish. You can say that of your own personal preferences. It's your preferences. Preferences can't be wrong. And that is where the problem lies. It's an answer that can't be wrong.
I think that's the litmus test for an helpful subjective answer: while the answer is spot on, it could have been wrong, if the author had a poor grasp of the subject.
Opinions can make very good answers, if they care thoughtful analysises. Strategies often are subjective in nature, yet are extremely useful answers. The answer to How do I know when it's time to push? is subjective in nature, but it can also be wrong. Pushing at level 1 is a very bad idea.