Moonlight is no longer thought of as the cause. In the 16th century, moonmilk, a soft white limestone precipitate found in caves, was thought to be caused by the rays of the Moon.
What I learned is that what makes it white is the reflection of the sun and the sun makes it glow, that's why it looks white.
And if you focus on the moon when it's not full you'll see the other part of it showing but it looks grey not white, because as I told you it's the suns reflection.
