  Fuzzy dark matter (FDM) is a dark matter candidate consisting of ultra-light
scalar particles with masses around $10^{-22} \mathrm{eV}/c^2$, a regime where
cold bosonic matter behaves as a collective wave rather than individual
particles. It has increasingly attracted attention due to its rich
phenomenology on astrophysical scales, with implications for the small-scale
tensions present within the standard cosmological model, $\Lambda$CDM. Although
constraints on FDM are accumulating in many different contexts, very few have
been verified by self-consistent numerical simulations. We present new large
numerical simulations of cosmic structure formation with FDM, solving the full
Schr\"odinger-Poisson (SP) equations using the AxiREPO code, which implements a
pseudo-spectral numerical method. Combined with our previous simulations, they
allow us to draw a four-way comparison of matter clustering, contrasting
results (such as power spectra) for each combination of initial conditions (FDM
vs. CDM) and dynamics (SP vs. $N$-body). By disentangling the impact of initial
conditions and non-linear dynamics, we can gauge the validity of approximate
methods used in previous works, such as ordinary $N$-body simulations with an
FDM initial power spectrum. Due to the comparatively large volume achieved in
our FDM simulations, we are able to measure the FDM halo mass function from
full wave simulations for the first time, and compare to previous results
obtained using analytic or approximate approaches. We find that, due to the
cut-off of small-scale power in the FDM power spectrum, haloes are linked via
continuous, smooth, and dense filaments throughout the entire simulation volume
(unlike for the standard $\Lambda$CDM power spectrum), posing significant
challenges for reliably identifying haloes. We also investigate the density
profiles of these filaments and compare to their CDM counterparts.
