We have recently developed a post-processing framework to estimate the abundance of atomic and molecular hydrogen (H I and H-2, respectively) in galaxies in large-volume cosmological simulations. Here we compare the HI and H-2 content of IllustrisTNG galaxies to observations. We mostly restrict this comparison to z approximate to 0 and consider six observational metrics: the overall abundance of HI and H-2, their mass functions, gas fractions as a function of stellar mass, the correlation between H-2 and star formation rate, the spatial distribution of gas, and the correlation between gas content and morphology. We find generally good agreement between simulations and observations, particularly for the gas fractions and the HI mass-size relation. The H-2 mass correlates with star formation rate as expected, revealing an almost constant depletion time that evolves up to z = 2 as observed. However, we also discover a number of tensions with varying degrees of significance, including an overestimate of the total neutral gas abundance at z = 0 by about a factor of 2 and a possible excess of satellites with no or very little neutral gas. These conclusions are robust to the modelling of the HI/H-2 transition. In terms of their neutral gas properties, the IllustrisTNG simulations represent an enormous improvement over the original Illustris run. All data used in this paper are publicly available as part of the IllustrisTNG data release.