I usually use these two terms to distinguish circumstances about which one may reasonably draw a variety of different, even conflicting conclusions (subjective) from circumstances about which there is a significant limit on the number or types of conclusions one can make. The easiest example may be a preference about food or art contrasted with a mathematical equation or the mass of an object. The first sorts of things can be disputed without a “right” or “wrong” answer (generally). The second category of things really can’t accommodate mutually exclusive conclusions; an object can only have one mass, two numbers can only sum to a single value.
The question does not seem to be whether these categories exist (though that may be disputed), but rather which types of circumstances fall into which categories. Can an explanation of a concept be objectively good or bad, or is the good explanation the one that conveys meaning and results in understanding? (Can someone be objectively good or bad at teaching?)
Another issue arises from this categorical dichotomy: how can we deal normatively with the subjective? It is easy enough to enforce objective laws and reward objectively positive performance—but how can we pass judgment on the subjective? The more circumstances we consider subjective, the less our tools of the modern, industrial era seem equipped to handle the world.
These questions apply readily to most issues related to art. It is often said that “art is subjective,” yet there are some things we are more comfortable considering art than others (a still life by a Dutch master vs. a pile of dog poo on a photo of a pop star). Further normative questions arise for art when we become concerned about untoward implications of the medium: Do violent movies, music, and games make us more violent as a culture? Although it is easy enough to make to the judgment to the effect that “a violent society is bad,” it is more of a challenge to say “this particular art ought not be published because of its possible results.” While issues of violence and suffering seem to be almost entirely objectively bad (except perhaps to a sadist), art remains subjective even while it depicts violence and suffering. Many focus on the question of whether such depictions cause mirroring realities, but I think there is also a question of what the depiction itself says. Art is a sort of text, and texts (may or may not) have intentions, goals, and meaning. The trend towards seeing more of the world in increasingly subjective ways has freed art in one way: to give it more flexibility and fluidity, particularly in interpretation. However, it may have restricted art in another way: by making it less clear what (if anything) a work means, it is less clear that it ought not to be censored.
The more subjective art is, the greater the danger of legitimate arguments for its censorship. The more objective art is the less flexible and fluid its interpretations become, perhaps undermining a core purpose of its existence.