The bright full Moon in the sky makes it appears as if a large amount of light is reflected off of the Moon's surface. This is not true. In fact, the Moon's surface is not very reflective at all. On average, only 7% of the Sun's light that strikes the surface of the Moon is reflected. So why does the Moon appear so bright in the night sky? There are a couple of reasons. For starters, a large amount of sunlight strikes the surface of the Moon, so even 7% of that is still a considerable amount of light. The larger reason, however, is that you are contrasting the full Moon against a dark sky. It's that contrast that makes the Moon appear very bright. Compare a full Moon that rises as the Sun is setting to a full Moon a couple of hours later. The same amount of sunlight is reflecting off the Moon's surface, yet the full Moon when the Sun is just setting does not appear nearly as bright as the full Moon an hour later when it is dark outside. This is due to the contrast of the Moon on a dark sky.
The Moon could appear MUCH brighter if there was a different surface composition. If the Moon had a very icy surface like Pluto and other far away objects in our Solar System objects, the amount of sunlight reflecting off would be MUCH greater than 7%. Europa, a moon of Jupiter's, and an object in our Solar System with a very icy surface, has a reflectivity of 64%. Imagine if our Moon had that reflectivity! Wow!!!