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Are pulsed neutron sources beams of neutrons that are not created continuously? If so what is the benefit of having a discontinuous beam of neutrons rather than a continuous one? Under what conditions is it better to use a neutron source in a particle accelerator than it is a proton beam?

Pulsed neutron sources are typically either nuclear reactors or "spallation sources."
At a reactor (a continuous source), the pulsing must be achieved by rotating some kind of absorber (a "chopper") in and out of the neutron beam.
At a spallation source, an energetic charged-particle accelerator is dumped into a heavy-metal target. The nuclei in the metal basically boil and emit lots of nasty things; the neutrons and photons are the longest-lived electrically neutral component, so they're what you see outside of the innermost shielding. You can isolate the neutrons from the photons by bending the beam around a corner.

Most high-intensity neutron sources (pulsed or continuous) are used for materials science, rather than particle physics. People (like me) who are interested in the free neutron from the perspective of nuclear and particle physics tend to set up experiments at facilities that are mostly producing neutrons for materials science purposes. Dubbers has written several review papers with titles like "doing particle physics with neutrons" which would interest you.

What makes neutrons nice from a materials-science standpoint is a kind of happy coincidence: a neutron from near the intensity peak of a room-temperature Maxwell-Boltzmann distribution has kinetic energy
$$frac12mv^2 approx kT = 25 text{ milli-eV.}$$

The de Broglie wavelength of such a neutron is

$$
lambda = frac hp approx 2,Å,
$$

which is not very different from the lattice spacing in a typical material.
So suppose you have a nuclear reactor for other purposes, like power generation, and part of the cooling systems for this reactor is a liquid water moderator. If you open a window in the shielding around this moderator so that some of the neutrons can escape, and put some material in the path of this cool neutron beam, the neutrons will undergo strong diffraction in a way that depends on the crystal structure in the material. If you have a cryogenic neutron moderator, rather than a room-temperature moderator, the neutrons have on average longer wavelengths and the diffraction effects are stronger.

Reactor neutrons are produced continuously while the reactor is operating. However, if you can make it so that your neutrons are produced in brief pulses, and put a flight path of length $L$ between the pulsed neutron source and your experiment, then you develop a simple relationship between the time of the neutron pulse, the arrival time of the neutron at your experiment, and the neutron wavelength:

$$
lambda = frac hp = frac h{mv} = frac h{mL} t
$$

As a grad student designing neutron flight paths, it became useful for me to memorize that the neutron has $h/m = 4.0 text{ Å m/ms}$. (It's actually 1% smaller than that.) That means, if your data acquisition clock starts when your pulse of neutrons is created, and your experiment is at the end of a flight path with $L=20,text m$, the 1Å neutrons arrive at $rm 5,ms$, the 2Å neutrons at $rm 10,ms$, the 4Å neutrons at $rm 20,ms$, and so on. In most countries the AC power from the electric grid provides a pretty good clock that's a few tens of milliseconds long, with which it's easy to synchronize your neutron pulses. So you can put a neutron detector at the end of your beamline, read off the neutron intensity as a function of time, and just cross off the "time" axis on your plot and write "wavelength" instead. It's a nice system.

From a particle-physics standpoint, neutron beams are completely different from proton beams. Neutrons, unlike protons, can't be re-accelerated. Neutrons can (in practice) only be steered by interacting with material neutron guides, where protons can be steered using electromagnetic fields. And neutrons have a nasty habit of going around corners, and of inducing radioactivity ("activation") in mostly everything that they land on. A person who cut their teeth doing particle physics at a proton accelerator would be instantly right at home doing particle physics at an electron accelerator; that same person would have a lot of "facts" to un-learn during their first year on a pulsed-neutron experiment.

I am not sure neutron beams are useful in particle accelerators, but they are useful in diffraction experiments, as one can determine the neutron velocity based on the time of flight.