What makes the difference on partially and fully visible moon?

In a word: shadows.

I cannot understand why the IQ is extremely diminished when doing the same with an almost fully visible moon.

The second image does appear to suffer from lower sharpness and overall quality. However, even if the technical image quality factors were equal, most importantly, a full moon will appear flat and uninteresting, as compared to a gibbous (i.e., 3/4-ish) moon.

We see a full moon that is lit directly "overhead". The mountain and crater rims do not cast any shadows that give texture and depth to the moon's surface. Thus, we only get tonal information from the albedo (reflectivity) of the local regolith in parts of the image.

However, with a partial moon, the surface is more side-lit, thus casting shadows. These shadows provide vital depth clues to our eyes, and greater tonal variations. Even if the technical image quality is the same (i.e., same accurate focusing, correct exposure, no motion blur, etc.), a partial moon's greater tonal and texture variations will make for a more apparently higher-quality moon image.

Your images provide for excellent comparison. Looking at Tycho Crater (the large impact crater in the south-southeast view, with large whitish ejecta streaks emanating from it): Notice in the gibbous phase image, Tycho crater has a distinct rim and bowl, and is surrounded by lots of smaller impact craters. Whereas, in the full moon image, the shape of Tycho crater is apparent, but it has no depth, no sense of being an obvious bowl. The smaller impact craters immediately surrounding it are nearly invisible, and the general area surrounding Tycho is just a smudge of middle gray.

The third image has substantially higher quality than the second image. However, even with its increased clarity and dynamic range, notice that the area immediately surrounding Tycho crater still doesn't exhibit much depth. The crater rim has more definition than the second image, but to my eye, doesn't exhibit nearly the same dimensionality and character as the first image does.