Why does the Moon look quite bright from Earth when it is full, but quite dim and faint from space?

The premise of your question is incorrect - why do you think the moon is dimmer when viewed from space? The brightness of the moon is a function only of how much sun is shining on the side that is facing you. Obviously the amount of light reflected off the surface toward your eyes is greatest when the moon's pahase is full. It doesn't matter whether you are standing on the earth or in a space capsule - if the moon is full it will be equally bright, unless your space ship ship is further away from the moon than the earth is (about 250,000 miles).

There is an added dimension to this, and it is the ~100 miles of atmosphere that is between you and the moon, as well. The Moon is, on average, about 240,000 miles away. That's about ten times a trip around the Earth's equator.

When the sunlight reflects off the Moon and heads to the Earth, it passes through our atmosphere. If the weather is right, that air can have a magnifying, spreading, or diffusing effect that makes the Moon appear bigger to your brain. Your brain then compares the apparent size of the Moon at 240,000 miles away, to the sizes of trees on the horizon, maybe 10 miles away at most, and thinks it knows how big the Moon is based on things you normally can touch.

Since the Moon is 24,000 times farther away than the tree, and much much much much bigger to begin with, your brain is fooled into thinking that the Moon looks bigger than it is. And then another part of the brain calculates the apparent brightness and basically does it incorrectly. Again, this is because your brain has no real value of how bright the Moon really is, it only knows how bright things are compared to each other.

Combine this with the atmospheric distortion, and your brain basically calculates an apparent brightness of the Moon as something more than what it would be if someone in space, 240,000 miles on the other side of the Moon, were to try to calculate the same brightness when the Moon was full from their perspective.