We present the first hydrodynamical simulations of structure formation using the new moving mesh code arepo and compare the results with gadget simulations based on a traditional smoothed particle hydrodynamics (SPH) technique. The two codes share the same Tree-PM gravity solver and include identical subresolution physics for the treatment of star formation, but employ different methods to solve the equations of hydrodynamics. This allows us to assess the impact of hydro-solver uncertainties on the results of cosmological studies of galaxy formation. In this paper, we focus on predictions for global baryon statistics, such as the cosmic star formation rate density, after we introduce our simulation suite and numerical methods. Properties of individual galaxies and haloes are examined by Keres et al., while a third paper by Sijacki et al. uses idealized simulations to analyse in more detail the differences between the hydrodynamical schemes. We find that the global baryon statistics differ significantly between the two simulation approaches. arepo shows systematically higher star formation rates at late times, lower mean temperatures averaged over the simulation volume and different gas mass fractions in characteristic phases of the intergalactic medium, in particular a reduced amount of hot gas. Although both hydrodynamical codes use the same implementation of cooling and yield an identical dark matter halo mass function, more gas cools out of haloes in arepo compared with gadget towards low redshifts, which results in corresponding differences in the late-time star formation rates of galaxies. We show that this difference is caused by a higher heating rate with SPH in the outer parts of haloes, owing to viscous dissipation of SPH's inherent sonic velocity noise and SPH's efficient damping of subsonic turbulence injected in the halo infall region, and because of a higher efficiency of gas stripping in arepo. As a result of such differences, arepo leads also to more disc-like morphologies in the moving mesh calculation compared to gadget. Our results hence indicate that inaccuracies in hydrodynamic solvers can lead to comparatively large systematic differences even at the level of predictions for the global state of baryons in the universe.