Basically, doubts are being raised about the previous suggestion that the Moon once had large quantities of water.
I wasn't even aware of this ever being a suggestion. A dry, volatile-poor moon would be a natural result of the Giant Impact, which has been pretty much the mainstream explanation for the Moon's origin since the Apollo age. Any appreciable water that the Moon does have was delivered by impacts after its formation.
The origin of Earth's water is slightly more mysterious -- it's not clear how much water was incorporated into the Earth during its accretion. Perhaps very little, as it formed inside of the 'snow line' of the Solar system, the distance at which water can form solid grains. By D/H ratios, it is thought that a good portion of the water came from comets. Water within the mantle got there by tectonics -- plate subduction and mantle convection.
This actually explains things a bit more clearly, as the water seems to have got there through tectonic means, and therefore probably arrived after the Earth had cooled somewhat. Which brings us back to the 'Comet Bombardment' theory caused by (possibly) Neptune taking a path through the Oort cloud a few billion years ago.
Precisely.

This is the
Nice model (named for the city in France, not because the model is so awesome), which explains the Late Heavy Bombardment due to planetary migration. It paints a very complex, dynamic evolution of the young solar system, and is widely accepted today amongst planetary scientists.