Phases: Not a Coincidence
The keen observer will note that the full Moon occurs only at night and that you'll never see a new Moon at midnight, this is because of the phases and orbit of the Moon. The phases of the Moon are a term used to describe the position of the Moon, relative to the Sun and Earth, where the Moon shows a certain amount of its reflected light to the Earth. What this means is that, as the Moon orbits the Earth, it exposes different portions of its body to the light from the Sun and reflects that light to observers on the Earth. When the Moon is halfway through an orbit of the Earth it will appear as a quarter Moon, with one half lit up by the light of the Sun and the other half dark. This appearance alters precisely with the position of the Moon around the Earth.
When the Moon appears in the daytime, it means that it's at the stage of its orbit where it is closer to the Sun than the Earth. While the Sun is incredibly bright, it's overall brightness isn't enough to completely wash out the Moon during the daytime, that is, when the Moon isn't too close to the Sun in the sky.
The difference between a new and full Moon is which side of the Earth, relative to the Sun, it's on. When the Moon is full, all of the light from the Sun will be reflected off the Moon and be visible to Earth, but when the Moon is new, none of that light is visible because the Moon is between the Earth and the Sun. Consequently, even when the Moon isn't exactly between the Earth and Sun, it still isn't visible, unless it's eclipsing the Sun. This is because of the glare of the Sun which washes out the minimal light being reflected from the Moon.