The colors of most things don't appear to change when you move from sunlight to fluorescent light to incandescent light to firelight, even though the actual ratios of wavelengths that are hitting your retina are quite different. This is because your brain automatically adjusts colors based on the ambient light. Since the moon is providing the ambient light, your mind automatically adjusts its color to be close to white.
So if you try to answer this question using physics alone (like some of the other answers do), you're likely to get the answer wrong; the perceived color of the moon depends on a combination of physics and neuroscience. See this question in biology.stackexchange.
In fact, you would expect the moon to be the same color as the sun, because both of these colors are produced by having the blue light refracted by the atmosphere. So why aren't they. So why aren't they? I would guess that the eye is not good at perceiving the moonlight scattered by the sky and off objects; it's so dim it just stimulates our rods and not our cones. Thus, when the color of the ambient light is averaged over by our brain, it is very close to the actual light coming from the moon, so the moon looks white. This explanation would predict that a bright full moon would look yellower than a half-moon. I believe it does.