Just as day is brighter than night, the sun is much brighter than the moon — 400,000 times brighter, to be exact. That's compared to even the fullest, brightest full moon. From an astronomical standpoint, this is no surprise. The moon doesn't generate its own light, and the burning sun provides the vast majority of all the natural light on earth.
Moonlight consists mostly of reflected sunlight. The moon even reflects some of earth's own light back. As anyone who has observed moonlight at various points of the moon's phase cycle will know, the moon shines brighter when it is full.