Ultra-diffuse galaxies (UDGs) are unusual galaxies with low luminosities, similar to classical dwarf galaxies, but with sizes up to similar to 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 Hubble Space Telescope (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 similar to 40% in the cluster core, where it is slightly lower than the nucleation fraction of classical dwarfs, to less than or similar to 20% in the outskirts. Collectively, we observe an unmistakable diversity in the abundance of GCs, and this may point to multiple formation routes.