It's not the Moon, it's your eyes.
I don't mean your eyes specifically, I mean all human eyes.
Human eyes have two different kinds of "sensors". As someone already mentioned, these are called rods and cones. Rods are only sensitive to the level of light and not the specific color. If you only had rods you would see everything in black and white and shades of gray. Cones are the sensors for color. It turns out that humans have a lot more rods than cones, which means (among other things) that it takes a lot more light to get the cones to work so we can perceive color. So, when there's a lot less light than normal, we see "black and white" much better than color.
Now take the case of the Moon. The Moon is a bit more than 1/4 the diameter of the earth, actually 27.3%. That means the Moon only gets about 7.5% as much light as the Earth (0.273^2=0.0745, please forgive my simplification here). Also, the moon is nearly black in color (really) and only reflects about 7% of the light hitting it.
In short, the moonlight is a whole lot less bright than sunshine, and that's just not enough to stimulate the cones very much.