Why are neutron scattering factors essentially independent of $2\theta$, whereas X-ray scattering factors drop off with increasing $2\theta$ ?
Scattering factors describe scattering of an atom
To predict how an object will diffract, you essentially have to do a 3D Fourier transform of the scattering density (https://chemistry.stackexchange.com/a/115815/72973). The scattering factor is derived from the scattering of a single atom (placed at the origin). For xray diffraction, the relevant density is that of the electrons. For neutron diffraction, it is that of the nucleus. See slide 11, https://www.embl-hamburg.de/biosaxs/courses/embo2012/slides/neutron-scattering-basics-trewhella.pdf
The fuzzier the scattering density, the higher the angular dependency
The Fourier transform of a Gaussian is a Gaussian again; a sharp Gaussian results in a broad transform, and vice versa. For X-ray diffraction, the electron density is fuzzy (same magnitude as the wavelength), so there is a large angular dependence. For neutron diffraction, the scattering density is very compact, leading to almost no angular dependence of the scattering factors.