Back when I was an undergrad in astrophysics one of my professors claimed that neutron stars are the most difficult objects to model in the universe. I believe him. In order to have a good model of a neutron star you need to use pretty much every branch of physics: general relativity, quantum field theory (both QED and QCD), fluid dynamics, and statistical mechanics. And these fields have important open questions in exactly the domains that you need to apply them. So you end up needing to solve not just one open problem, but a whole bundle of them, and they all conspire to work together to make each one harder than it would be on its own.

Consequently, not very much is actually known about neutron stars. One of the most important open problems in modeling neutron stars is just coming up with the most basic relationship you can, namely the "equation of state." This is simply a relationship between the density and the pressure within the neutron star. The equation of state of normal stars was effectively developed in the 19th century with the development of statistical mechanics. The equation of state of white dwarfs was developed by Chandrasekhar in the 1930s, not long after the development of quantum mechanics (and while Chandrasekhar was on a boat traveling from India to England!). But the equation of state for neutron stars is still unknown.

A major consequence of the equation of state is that it predicts the maximum mass of the object. In the case of a white dwarf, Chandrasekhar showed that the equation of state implied that the maximum mass had to be about 1.44 times the mass of the Sun (this is now appropriately known as the Chandrasekhar limit). But because the equation of state of neutron stars is not known, it is not known what the maximum mass of a neutron star is. Therefore, discovering more massive neutron stars is very important because it can possibly rule out certain equations of state. This, in turn, tells us a little bit more about what physical processes are going on in the neutron star.

Since the problem is so theoretically intractable astrophysicists usually have to neglect or simplify certain aspects of the physics considerably. If the equation of state that comes out of these assumptions can be ruled out, this means that one of these assumptions must be faulty, and some of the underlying physics that was neglected turns out to be important after all.