We investigate the impact of the number of filaments connected to the nodes of the cosmic web on the physical properties of their galaxies using the Sloan Digital Sky Survey. We compare these measurements to the cosmological hydrodynamical simulations H？ORIZON-(NO)AGN and SIMBA. We find that more massive galaxies are more connected, in qualitative agreement with theoretical predictions and measurements in dark-matter-only simulations. The star formation activity and morphology of observed galaxies both display some dependence on the connectivity of the cosmic web at a fixed stellar mass: Less star forming and less rotation supported galaxies also tend to have higher connectivity. These results qualitatively hold both for observed and for virtual galaxies, and can be understood given that the cosmic web is the main source of fuel for galaxy growth. The simulations show the same trends at a fixed halo mass, suggesting that the geometry of filamentary infall impacts galaxy properties beyond the depth of the local potential well. Based on simulations, it is also found that active galactic nucleus feedback is key to reversing the relationship between stellar mass and connectivity at a fixed halo mass. Technically, connectivity is a practical observational proxy for past and present accretion (minor mergers or diffuse infall).