Short answer: the vast of majority of water delivered by comets to the moon was lost to space. Actually, some of the water on earth has escaped and is escaping to space, too, but the Earth loses water much much more slowly than does the moon.
The moon loses water rapidly because:
1) Its gravity is only one tenth of earth's gravity, since the moon is much less massive than the earth, while its surface temperature is about the same as that of earth, since the earth and moon are the same distance away from the sun. Temperature is essentially a measure of the average speed of the molecules in a substance. Technically speaking, temperature is a measure of the average kinetic energy of the molecules. Kinetic energy is equal to mass times velocity squared (m * v^2), so the lighter a molecule is, the faster it's average speed. Water molecules (in gaseous form) at typical lunar temperatures have, on average, enough speed to escape the moon's weak gravity. This process is sometimes referred to as thermal escape. On earth, the average water molecule has far too little speed to escape from the earth's relatively strong gravity, but there will be a few molecules that do have sufficient speed.
2) The moon as no atmosphere. Why does the moon have no atmosphere? Basically, because the previous argument applies to any of the molecules that might be available to form a lunar atmosphere. Both liquid water and ice are unstable when there is little or no atmospheric pressure - they will evaporate or sublimate, respectively, to become water vapor, which can then be lost to space via thermal escape. On earth, because there is plenty of atmospheric pressure (and because the temperature is not too high) the vast majority of water vapor becomes liquid or solid again, forming clouds, before it can make it to the upper parts of the atmosphere where it can be lost by thermal escape. (In the lower atmosphere on earth, fast moving molecules will collide with other molecules before they have a chance to escape.)
3) Ultraviolet light from the sun breaks water molecules apart into hydrogen and oxygen atoms. This process is known as "photo-dissociation". Since the hydrogen atoms are much lighter than oxygen atoms, the hydrogen atoms are lost to space much more rapidly than the oxygen. On earth, the ozone layer shields the lower part of the earth's atmosphere from ultraviolet light, thereby protecting water molecules from destruction. (The particles of the solar wind also collide with water molecules and destroy them, just as ultraviolet photons do. Conveniently, the earth's magnetic field deflects the solar wind.)
Actually, the retention of water by the earth is the exception rather than the rule in the inner solar system. The planet mercury, like the moon, lost the vast majority of water delivered by comets, and for much the same reasons. Venus, on the other hand, lost most of its water because its atmosphere, while plenty thick enough, was too warm, allowing water vapor to hang around in its upper atmosphere where it was lost to photo-dissociation, thermal escape, and the solar wind.
Although Mercury and the Moon lost MOST of water that was brought to them by comets, they might not have lost all of it. If some water molecules end up in an exceptionally cold spot on the surface of the Moon or Mercury (a crater near the pole that never receives direct sunlight, for example) then the sublimation of those water molecules will proceed very slowly, just as all chemical reaction do when the temperature is very low. In the case of Mercury there is good evidence that some water is in fact retained in exactly this manner. The telltale radar signature of ice has been observed in regions that just so happen to correspond with craters that never receive direct sunlight. The evidence for water-ice on the moon is less convincing. A reported radar detection by the Clementine spacecraft has been disputed and has never been duplicated. The Lunar Prospector spacecraft showed that there was extra hydrogen at the moon's pole. Extra hydrogen suggests, but does not prove, the presence of water.