Is there anything different about the gravitation around a non-spinning black hole and a neutron star of the same mass?
Birkhoff's theorem is very useful: in general relativity, if you are in vacuum and there is a spherically symmetric gravitational field, then it will be the Schwarzschild solution. This solution only depends on the mass, not on the size of the object. So the neutron star and the black hole will give rise to exactly the same orbits.