Yes and no. It's a fact that 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. However, their difference lies on their parent star. It's best to compare black holes and neutron stars 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 that's small yet has a mass which is 1.4 to 3 times that of the sun, it will form a neutron star. Once formed, they no longer actively generate heat, and cool over time. A neutron star has some of the properties of an atomic nucleus, including density and being composed of nucleons. Stars that has a mass greater than thrice the sun's mass, a black hole is formed, having only three independent physical properties: mass, charge and angular momentum. The black hole otherwise has no features.