The sky is blue (I'm told) because nitrogen in the atmosphere scatters short blue wave lengths of light from the sunlight, which is also why the sun appears somewhat yellowish rather than white, as it does in space. I know that the moon is not bright enough to make the sky blue but shouldn't the 'blue' part of it's spectrum still be filtered out of its image regardless making it more yellow or red in colour?
Moonlight is actually slightly redder than sunlight, because the reflectance spectrum of the full Moon is higher in the red part of the spectrum than the blue.
The effect of the atmosphere (which of course is exactly the same for sunlight and moonlight) is not very large when looking at both objects when they are well above the horizon. If both were overhead then something like 50% of the bluest light you can possibly perceive (maybe 350 nm) is scattered/absorbed compared to about 20% at 550nm and perhaps 15% at 700nm. This sounds like a lot, but the eye/brain does not perceive colour in a way that is linearly proportional to the flux received at any wavelength and this level of modification is perhaps equivalent to changing the perceived colour from that of a G2 star to a G5 star (not a big change -- see here). It is not until the Sun gets very low towards the horizon that there is a very noticeable change in colour, and as I say, this affects moonlight in exactly the same way.
I think if you are claiming that the Sun and Moon appear to be different colours then I won't argue. But it is a question of perception for two objects whose brightness is different by many orders of magnitude and therefore the answer probably lies in the physiology of the eye/brain and not Physics.
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? 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.
To me the moon appears yellow enough but here are its true colours: http://www.olino.org/us/articles/2015/10/05/spectrum-of-moon-light
We need to start with the fact that white light is a combination of all colors produced equally by a glowing object. A glowing object that appears blue is blue because it's producing more blue light than it is producing red, orange, yellow, green light. The color of a glowing object depends on the temperature of the object. Now we can proceed to your question. Two Reasons why the Sun appears yellow:
- The Sun's surface temperature (5,500 degrees C) produces a range of visible light (red to blue) in which yellow is the most plentiful, but not much more than other colors it produces. If the Sun were cooler, say 2,500 degrees C, it would look red, like the stars Antares and Betelgeuse. Or if the Sun were hotter, say 15,000 degrees C, it would look blue, like the star Rigel.
The Earth's atmosphere acts as a kind of light filter. Some colors are filtered more than others. The Sun is a yellow star, but the Earth's atmosphere makes the Sun look more yellow than it appears than if you were to observe it from space where it would appear more white than yellow. But you don't have to leave Earth to see that the Sun is really less yellow than it appears. If you are in the Rocky Mountains at 11,000 ft elevation, the Sun looks less yellow and more white than it does at sea level. There are fewer air molecules at this elevation to filter the Sun's other colors. Imagine what the Sun would look like from an airplane at 40,000 ft altitude--quite white! Also, when you are able to look at the Sun where you live, it's morning or late afternoon. It's easier to look at the Sun for a few seconds than it is a noon. The Sun appears more yellow at those times than it would if you were to observe it at noon (12 PM) when Sun is highest in the sky for the day; it's at its brightest and whitest--hard to look at. Because of the Sun's high position at noon, the sunlight has less air to travel through. Less air means less filtering of other colors. Remember: Light appears white because all colors are equally reaching your eyes. So, at noon the Sun appears to be more white, less yellow--closer to the way it really is! (Don't try to make this observation without hi-tech eye protection).