A newly formed neutron star will become a pulsar if it is rotating fast enough and has a strong magnetic field- most probably do. Over time, the pulsar rotates more slowly as it radiates electromagnet energy. Eventually, there is no longer enough rotational energy to drive the pulsar.
Whenever a type 1A supernovae releases it's remnants, if the left over rigid core doesn't exceed TOV limit then in that case a neutron star is formed. I suppose the difference between a neutron star and a pulsar would be on the scale of sizes of the magnetic field as hinted by jimmy's post.
Pulsars accrete surrounding matter onto their surface. The strong magnetic field of the neutron star funnels the matter onto the poles of the star, and bright hotspots are created on the surface. If a neutron star has nothing to accrete, there are no hotspots, and there is no pulsar.