After a supernova, what factor determines whether a star becomes a black hole vs. a neutron star?
The fate of a supernova star core depends on its mass.
A supernova explosion occurs when the core of a large star is mainly iron and collapses under gravity. The pressure causes protons and electrons to combine into neutrons forming a neutron star. The energy released in the process blows away the outer layers of the star.
What happens next depends on the mass of the neutron star. If it is less than about three solar masses it remains as a neutron star.
If the neutron star is more than about three solar masses then the pressure exceeds the neutron degeneracy pressure. This causes the neutron star to collapse into a black hole.