  Ultra-diffuse galaxies (UDGs) are unusual galaxies with low luminosities,
similar to classical dwarf galaxies, but sizes up to $\sim\!5$ larger than
expected for their mass. Some UDGs have large populations of globular clusters
(GCs), something unexpected in galaxies with such low stellar density and mass.
We have carried out a comprehensive study of GCs in both UDGs and classical
dwarf galaxies at comparable stellar masses using HST observations of the Coma
cluster. We present new imaging for 33 Dragonfly UDGs with the largest
effective radii ($>2$ kpc), and additionally include 15 UDGs and 54 classical
dwarf galaxies from the HST/ACS Coma Treasury Survey and the literature. Out of
a total of 48 UDGs, 27 have statistically significant GC systems, and 11 have
candidate nuclear star clusters. The GC specific frequency ($S_N$) varies
dramatically, with the mean $S_N$ being higher for UDGs than for classical
dwarfs. At constant stellar mass, galaxies with larger sizes (or lower surface
brightnesses) have higher $S_N$, with the trend being stronger at higher
stellar mass. At lower stellar masses, UDGs tend to have higher $S_N$ when
closer to the center of the cluster, i.e., in denser environments. The fraction
of UDGs with a nuclear star cluster also depends on environment, varying from
$\sim\!40$\% in the cluster core, where it is slightly lower than the
nucleation fraction of classical dwarfs, to $\lesssim20\%$ in the outskirts.
Collectively, we observe an unmistakable diversity in the abundance of GCs, and
this may point to multiple formation routes.
