We explore the differences in gas-rich field ultra-diffuse galaxies (UDGs) and diffuse classical dwarf galaxies using an extensive atomic gas (H I) follow-up survey of optically selected UDG candidates from the Systematically Measuring Ultra-diffuse Galaxies (SMUDGes) catalogue. We also compare the SMUDGes-H I observations with two state-of-the-art cosmological hydrodynamical simulations: Numerical Investigation of a Hundred Astrophysical Objects (NIHAO), where UDGs form through a series of bursty star formation episodes and ROMULUS25, where UDGs form as a result of major mergers that temporarily increase their spin. Although the suggested formation scenarios for UDGs within these simulations are different, the present-day H I masses MH I, stellar masses M*, and star formation rates of simulated galaxies are qualitatively and quantitatively consistent with each other and with the observed SMUDGes-H I sample. We find that when controlling for M*, there is a positive correlation between the gas richness MH I/M* and the effective optical radius Reff, and that this trend is not different between the UDG and dwarf populations, within the measured scatter. Taken together, our results suggest that gas-rich, star-forming UDGs and dwarfs are not distinct galaxy populations, either observationally or in simulations.