Nothing in nature ever stays exactly the same, but some things happen faster than others.
Neutron stars are born in extraordinary physical circumstances, squeezed by shock-waves of infalling and superheating material as the core of a massive star gives way against inexorable forces of pressure and its own mass.
The result is that a young neutron star is born in the blaze of a supernova explosion, and emerges as a rapidly spinning (tens of times per second) and extremely hot surface (above a million degrees Kelvin).
The energy from both the spin and the internal temperature make these objects visible even at great distances, particularly in X rays and radio waves, for tens of thousands of years. But since neutron stars have no "new" nuclear reactions taking place to power them, they are simply lighthouse cinders banking on past glory. They fade and slow their spin, and eventually become essentially impossible to detect directly from the Earth. They aren't "dead" -- they retain all their mass, and may even be orbited by their companion stars (and in at least one case we know of, planets), but the spectacle fades.
That is, most of the time.
In the case of a neutron star orbited by a companion star at very close distances, the end of life of the companion star may make life interesting for the neutron star. Stars swell and cool towards the end of their life when hydrogen is exhausted in their cores. If the expansion of the companion star moves its surface sufficiently close to the neutron star, gas from the companion star will spill over and spin down into the gravity well of the neutron star, eventually accumulating on its surface. This not only reheats the neutron star, it spins it up, in fact producing spins faster than the spin with which the neutron star was born. We have measured neutron stars with spins exceeding 250 times per second in these binary systems.
If enough matter is transferred, the neutron star may become too heavy for the pressure of nucleonic repulsion to "hold up" the star, and a further collapse will then occur. The result: another explosion, and a final corpse consisting of a black hole.