Kang: Abortions for all! Crowd: Booo! Kang: All right, then. No abortions for anyone! Crowd: Booo! Kang: Hmm... Abortions for some, miniature American flags for others! Crowd: Yaaay! —The Simpsons

I don't understand how this is an example. "Keep abortions legal but discourage them in other ways" is a legitimate moderate position.

That's clearly not the reasoning behind the Simpsons example, which process is plainly evident in the dialogue.

With the sheep-painting example, a "true" Golden Mean would try its best to incorporate elements of painting both colors, eg, paint half the sheep's bodies one way and half the other, or mix the two colors and paint all the sheep with the result, etc. In real life situations, the truth may well lie between or outside two positions, though often one side or the other will be a little closer to the truth (at least if we're talking about something objectively determinable, eg, the age of the Earth).

The fundamentally fallacious thing about the Golden Mean is that it seeks correct answers in terms of what the existing positions are, rather than in terms of the facts on the ground (or whatever). The argument from popularity and the argument from authority make similar mistakes. It's also possible to invert the fllacy, eg, to argue that we shouldn't paint any sheep because "both sides" should be made to feel unhappy. In fact, a good rule of thumb is that the phrase "both sides" heralds a potential Golden Mean coming. (saying "Both sides have their points" usually doesn't add to the conversation, and is rarely followed up by "but one side is righter".)

This is late, but anyway… the difference is that you usually compromise out of the principle that if you want X, well, a little X is better than no X at all. The Golden Mean fallacy, conversely, holds that if the X and Y positions are such extremes, the middle position Z must be best because it is in the middle. The conclusion may be the same but the process is different.

Notice how this is also distinct from plain political moderation. One might happen to support position Z on its own merits, and this is different from believing that Z is great because it pleases everyone equally, and/or it pisses everyone off equally. The Klang and Kodos example is a good one in that it shows the process of a Golden Mean in the making, in contrast to a formation of a moderate view on its own merits.