What is the key observation needed to determine whether a compact object in question is a neutron star or a black hole?
The key factor which determines whether an object is a neutron star or a black hole is its mass.
Neutron stars and black holes have many similarities. They are degenerate and both formed when the iron core of a massive star collapses under gravity.
They are both small and massive and can be spinning and charged. Both can emit radiation.
The key to identifying whether an objects is a neutron star or a black hole is its mass. If has a mass less than about 3 solar masses it is probably a neutron star. If is more than 3 solar masses it is a black hole.
The reason for this is neutron degeneracy pressure. This is a quantum effect which keeps neutrons separate even under high temperatures and pressures. If a neutron star is more than 3 solar masses the gravitational pressure is so great that the star undergoes further collapse into a black hole.
An objects mass can easily be determined if something is orbiting around it. If the orbiting body's period and semi major axis distance can be determined, the mass of the object can be calculated