If they were spinning they would be distinguishable (in principle), otherwise not.
Astrophysical black holes and neutron stars are expected to spin. In the case of a neutron star that automatically means that the mass/energy distribution is not spherically symmetric and therefore that the detail of the potential outside the surface depends on the detail of how the mass is arranged (e.g. the mass quadrupole and the spin octopole), which in turn depends on the equation of state of the neutron star, not just on its mass and angular momentum - see for example Pappas & Apostolatos (2013); Pappas (2017); Frutos-Alfaro (2018) for calculations of the metric outside spinning neutron stars.
For a spinning black hole, the potential is determined by the Kerr metric and simply depends on the total mass and angular momentum with no additional details required.
i.e. The spacetime metrics outside a spinning black hole and a spinning neutron star with the same $J$ and $M$ are different.
The difference in the metrics has observable consequences (in principle) for say the innermost stable circular orbit (ISCO), the orbital frequency at the ISCO and for orbital precession. For example, see the plot below (from Luk & Lin 2018), which shows there are big differences in the ISCO radius if the value of $J$ is large (in practice, I think this means rotation frequencies of $\sim 1$ kHz are required).
For non-spinning objects, Birkhoff's theorem applies, as described by Anders Sandberg, and there would be no distinction.