From what I understand, a hydrogen bomb usually has neutron reflectors, like graphite, to increase the yield of the explosion. A neutron bomb does not have those reflectors. Are there any other differences to how they're built?
Graphite turns fast neutrons into slow ones. Since fission bombs run on fast neutrons, graphite is not used in them. Instead, all fission bombs have reflector casings called tampers which are made of dense metal, not graphite, with the intent of keeping the fast neutron density inside the fissionable material as high as possible for as long as possible and thereby reducing the amount of fissionable material required to constitute a critical mass.
The inertia of the dense metal reflector also helps confine the exploding bomb for a few extra microseconds before its expansion quenches the chain reaction.
In the case of a hydrogen bomb, the tremendous burst of neutrons produced when it explodes can then be used to trigger fission reactions in the bomb casing as the neutrons depart the scene. That casing is made of things like spent uranium reactor fuel and U-238. This extra fission stage allows gigantic energy release in a hydrogen bomb.
To make a neutron bomb, the bomb casing is made of non-fissionable material and so up to 40% of the energy release in this case is contained in the fast neutrons, while the blast effects due to heat and pressure from the bomb are minimized by design.