The angular concentration of the light. The moon is diffuse, so an incoming ray of sunshine is spread by the optically rough surface of the moon from an incide

The angular concentration of the light. The moon is diffuse, so an incoming ray of sunshine is spread by the optically rough surface of the moon from an incident solid angle of 6 * 10^-5 steradians out into 2 pi steradians of the night sky, or a reduction in angular concentration by a factor of about 100,000 (totaling ~1 million after the albedo is accounted for). It is like you are looking at the sun through a mirror so rough that the image of the sun is blurred over literally half the sky. Because this process does not create new photons, the blurred image must be far, far dimmer. This circumstance corresponds to the "most we could do" with lenses and mirrors focusing the moon, which is to fill the sky with an image of the moon/"blurred sun". Unconcentrated moonlight corresponds to the same picture, except we only see a "cutout" disk of this blurred sun-image which is the size of the moon in the sky. Our crappy moon-mirror does not fill our vision, it is a porthole letting through only a tiny fraction of the blurry sun-image. And of course if you imagine yourself as the ant under the magnifying glass, with your entire sky filled with moon, there is no way you could spontaneously become hotter than your moon-y surroundings.