Are there any known neutron stars just below the Tolman–Oppenheimer–Volkoff limit? Do we know any neutron stars just below the Tolman–Oppenheimer–Volkoff limit?
And how are they different from most other neutron stars?
 A: I assume you mean the Tolman-Oppenheimer-Volkoff limit (TOV limit). Proceeding accordingly:
The most massive neutron star yet found that I know of is 1.97 solar masses. It is presently unknown how close that is to the real TOV limit.
The value that one calculates/predicts for the TOV limit is dependent on the equation of state, i.e., the relationship between pressure and density, which for the incredibly dense matter found inside a neutron star, is still an open field of research. If your preferred EOS is relatively squishy, i.e. pressure is relatively small for a given density, then the neutron stars you calculate will be bad at supporting themselves, so the TOV limit in that EOS will be low. Demorest's very heavy neutron star eliminates the entire squishy end of the spectrum of EOS theories, which is kind of a bummer, actually.
A big factor in the stiffness of an EOS is how many different degrees of freedom matter has as it gets denser. For instance, if an exotic new phase of matter comes into play like strange matter, pion condensate, hyperon phases, or quark-gluon plasma, then at the transition, particles can be in both regular matter states and the exotic states, which acts to reduce stiffness (technically speaking, I am referring to density of states for Fermions). Since the real EOS is apparently very stiff, it makes a lot of these cool theories much less probable.
Demorest's neutron star is different from others only in its significance to the field of extreme EOS research, as far as I know. It's kind of the point of giving the TOV limit a name that everything below it is only quantitatively different, not qualitatively. Its binary system is also fortuitously aligned for observations taken from Planet Earth, but that's also not an intrinsic or physical thing.
A: Browsing the wikipedia article on Neutron Stars brings up a recent arXiv posting on the known mass distribution which suggests one possible hit (B1516+02B) at $2.10 \pm 0.19$ solar masses, the $1.97 \pm .04$ solar mass NS that Andrew writes about, and another  (see table 2) and another at $1.91$ solar masses.
Figure 3 is very interesting in that it shows the central values and one sigma bounds all together.
