We use a pair of high resolution N-body simulations implementing two dark matter models, namely the standard cold dark matter (CDM) cosmogony and a warm dark matter (WDM) alternative where the dark matter particle is a 1.5keV thermal relic. We combine these simulations with the GALFORM semi-analytical galaxy formation model in order to explore differences between the resulting galaxy populations. We use GALFORM model variants for CDM and WDM that result in the same z=0 galaxy stellar mass function by construction. We find that most of the studied galaxy properties have the same values in these two models, indicating that both dark matter scenarios match current observational data equally well. Even in under-dense regions, where discrepancies in structure formation between CDM and WDM are expected to be most pronounced, the galaxy properties are only slightly different. The only significant difference in the local universe we find is in the galaxy populations of "Local Volumes", regions of radius 1 to 8Mpc around simulated Milky Way analogues. In such regions our WDM model provides a better match to observed local galaxy number counts and is five times more likely than the CDM model to predict sub-regions within them that are as empty as the observed Local Void. Thus, a highly complete census of the Local Volume and future surveys of void regions could provide constraints on the nature of dark matter.