Both Black holes and neutron stars are product of a dying star. When a star dies, it spent all of its energy and then collapses. Their difference lies on their parent star. For the purpose of this discussion, we compare them with the mass of the sun. If a star similar to that of the sun's mass dies, it will form a white dwarf. When a dying star has a mass which is 1.4 to 3 times that of the sun, it will form a neutron star. Stars with a mass greater than thrice the sun's mass, black hole is formed.
Black hole: Has a density of 2x10^30 kg/m^3, has high gravitational field that even light cannot escape. This is also the reason why black holes are hard to find. Physical manifestation of the objects around them can be a proof that they exist.
Neutron star: Has a density of 3x10^17 kg/m^3.
Black Hole: An object with a gravitational field so strong, not even light can escape. Has a mass ranging anywhere from 1.5 solar masses (stellar-mass) to billions of solar masses (supermassive black holes). Now, the density varies just as much. An earth sized black hole would have a density around 2 x 10^30 kg/m^3, while a supermassive black hole (believed to be at the center of most galaxies) would have a density similar to that of water.
Neutron Star: Collapsed remnant of a supernova. Has a mass ranging from about 1.35 solar masses to 2.1 solar masses. They typically have densities varying from 8 x 10^13 g/cm^3 - 2 x 10^15 g/cm^3, or about the density of an atomic nucleus.