Why is a bright night sky dark?
Moonlight has a very similar spectrum to sunlight - it is slightly "pinker", but that isn't really relevant here.
However, the flux of light at the earth from the (full) moon is approximately 400,000 times less than from the Sun. This matters because the colour-sensitive receptor cells in your eye (known as cones) do not work well at low light levels.
If you actually measure the spectrum of the nighttime sky, produced by scattered moonlight, it is in fact quite similar to that of the daytime sky, but the lower light levels mean the bluish colour is not perceived.
The night sky under full moon does look dark blue (provided low light pollution). However, the scattered "blue light" is not strong enough so that the night sky is still transparent to many stars. Of course, it's difficult to see dim stars during full moon. You may have better perception when you compare the background of full moon with thin clouds.
During day time, the blue sky may still transparent to half moon (of course between Moon rise and Moon set). Moreover, you can see some bright stars just before Sun set or after Sun rise. Think about the varieties of colours of sky during morning, noon and night from East, zenith and to West.
See the answers from another forum here and also Wikipedia article of Midnight blue.