Why do some neutron stars become pulsars? I suspect this may be an open problem in astronomy, but I have to ask anyway.

There doesn't seem to be any large-scale difference in the formation of pulsars from the formation of "normal" neutron stars. Yet pulsars end up emitting electromagnetic radiation, while others - "radio quiet" neutron stars - do not.

Why do are some neutron stars formed as pulsars, while others are not?

This is partly based on What causes a star to become a pulsar?, but the answer does not cover this part.
My understanding is that it would be quite usual for the majority of neutron stars to be pulsars shortly after they formed.

The requirements are that they should have strong magnetic fields and rapid rotation (faster than a rotation period of a few seconds). The latter seems an almost inevitable consequence of conservation of angular momentum and the former must surely be assisted (if not fully explained) by conservation of magnetic flux as the core of a massive star collapses (see the discussion in Reisenegger 2007).

But to be observed as a pulsar, the spin and magnetic field axes have to be oriented in a specific way, so that Doppler-boosted and beamed radiation sweeps over the Earth. This requirement means that we can observe only a fraction of the pulsars out there.

In fact there are a category of "radio quiet pulsars" - these are detected by their high energy radiation pulsations but not seen as radio pulsars. The idea here is that the beaming is less extreme for the high energy radiation, allowing it to be seen.

The pulsar phenomenon is also transient. Once the rotational kinetic energy of the pulsar has been extracted and it spins down beyond the "pulsar death line", or if magnetic field decay occurs rapidly, then the pulsar should switch off. Most of the billion (or so) neutron stars in our Galaxy are no longer pulsars. The slope of the death line is such (e.g. see Arons 1999) that even if the pulsar is born with a relatively weak field ($10^{6}$ T !) it should still be born with a rotation period short enough ($&lt;1$ s) to produce the pulsar phenomenon initially.

You might find the discussion/simulations in this paper interesting.