In addition to the other answers, an important physical difference is the perceivable inconsistency between the apparent intensity of the light source (moon vs. sun), and the spectral (color) distribution.
Of course, this requires the (obviously wrong) conjecture, that both are thermal light sources, which behave approximately according to the black body radiation profile (Planck's law).
In colloquial terms: an object as big as the moon, if it were a thermal light source, that emits only such a small amount of light in the direction of the observer, would be much cooler than the sun (see Stefan-Boltzmann law), and hence, would have a lot more reddish tint to the color of its light. Within certain limits we are able to perceive different color temperatures, and possibly a plant could do that too, given appropriate cells or chemistry. Hence, as our brain would naively expect the moonlight to be much redder, if it were a thermal light source, this is why we can subjectively tell that the moon is not a thermal light source.
Whether the biological species you mention is capable of comparing intensity with color, this I do not know. And the big question is, what is the benefit of evolving such a complicated mechanism, when probably it is for the plant much biologically easier to compare intensity with a given threshold value.