Please note that water is composed by two hydrogen's and one oxygen. The stable isotope of oxygen has atomic number 8, but atomic mass 16, i.e it is composed out of eight protons and eight neutrons. Thus in normal water the ratio of hydrogen contribution to the over all in mass is 2/18 .
When deuterium takes the place of hydrogen we have heavy water, and that means that two more neutrons are added to the original mass balance, an effect again of 2/18 extra mass. So the water is more dense but by this small factor.
In this article the advantages of heavy water in reactors are given.
Using water as a moderator will absorb enough neutrons that there will be too few left over to react with the small amount of 235U in the fuel, again precluding criticality in natural uranium.
heavy water seems to solve the problem:
An alternative solution to the problem is to use a moderator that does not absorb neutrons as readily as water. In this case potentially all of the neutrons being released can be moderated and used in reactions with the 235U, in which case there is enough 235U in natural uranium to sustain criticality. One such moderator is heavy water, or deuterium-oxide.
The reason is that deuterium has already bound a neutron to the hydrogen atom, and thus less neutrons needed for criticality are absorbed.