The hydrostatic equilibrium between gravity and the internal kinetic energy (pressure) of a neutron star in General Relativity is governed by the Tolman-Oppenheimer-Volkoff equation. This is like the usual Newtonian hydrostatic equilibrium equation except that there are multiplicative terms on the right hand side that increase the required pressure gradient as the pressure of the fluid increases and as the curvature of space increases.
What this means is that as you increase the central pressure of the neutron star, the pressure gradient required to support it becomes even larger. That is because the kinetic energy of the neutrons (or whatever else is in the interior) is actually contributing to the mass-energy that is curving space. This eventually becomes self-defeating - an additional increase in the central pressure to provide an increased pressure gradient just increases the required pressure gradient by even more! The star becomes unstable and collapses.
This imposes a limit on $M/R$ and for a given equation of state implies a maximum mass for the neutron star -- the Tolman-Oppenheimer-Volkoff limit.
The maximum compactness of a neutron star can be estimated in various ways. e.g. on p.260-261 of "Black Holes, White Dwarfs and Neutron Stars" (Shapiro & Teukolsky) it is shown that if causality is satisfied and the speed of sound is less than the speed of light, then for a stable neutron star, $GM/Rc^2 < 0.405$, which means that $R > 1.23R_s$, where $R_s$ is the Schwarzschild radius. Even if you abandon causality, it is easy to show from the TOV equation itself that even with infinite central pressure, that $GM/Rc^2 < 8/9$ and hence $R>1.13 R_s$.
Thus the radii of stable neutron stars are always somewhat above the Schwarzschild radius of a black hole with the same mass. The reason that the radius at which they become unstable is similar to $R_s$ is just that this is the size scale at which the GR effects that cause the instability become very important. It is not that degeneracy pressure "fails", it is that no stable solution is possible in GR.
Note that this final conclusion about the minimum radius as a function of $R_s$ is almost independent of assumptions about what provides the equation of state, but the corresponding mass at which this compactness limit is reached is sensitive to the exact equation of state. Neutron stars are not supported by ideal neutron degeneracy pressure, they are mainly supported by repulsive strong nuclear forces between closely packed neutrons. This allows them to achieve maximum masses of at least $2.1M_{\odot}$ and still satisfy the compactness limits above. If neutron stars were supported by ideal neutron degeneracy pressure than no neutron stars would exist with $M>0.75 M_{\odot}$, as was shown by Oppenheimer & Volkoff in 1939.
What is the "in between" state of matter at even higher densities during the collapse to a black hole? Nobody knows. It is quite likely that there are new "hadronic phases", which basically means the kinetic energy of neutrons gets turned into the rest mass of heavy hadrons like $\Lambda$ and $\Sigma$ particles. It is also possible that at very high densities (maybe 10 times that of nuclear matter), the quarks attain "asymptotic freedom" and there is a new phase of matter consisting of effectively free up, down and strange quarks.
