We present a detailed comparison between tidal effective-one-body (EOB) models and new state-of-the-art numerical relativity simulations for nonspinning binary neutron star systems. This comparison is the most extensive one to date, covering a wide range in the parameter space and encompassing the energetics of the binary, the periastron advance, the time and frequency evolution of the gravitational wave phase for the dominant mode, and several subdominant modes. We consider different EOB models with tidal effects that have been proposed, including the model with dynamical tides of Hinderer et al. Phys. Rev. Lett. 116, 181101 (2016) and the gravitational self-force (GSF) inspired tidal EOB model of Bernuzzi et al. Phys. Rev. Lett. 114, 161103 (2015). The EOB model with dynamical tides leads to the best representation of the systems considered here; however, the differences to the GSF-inspired model are small. A common feature is that for systems where matter effects are large, i.e., stiff equations of state or small total masses, all EOB models underestimate the tidal effects and differences to the results from numerical relativity simulations become noticeable near the merger. We analyze this regime to diagnose the shortcomings of the models in the late inspiral, where the two neutron stars are no longer isolated bodies moving in vacuum. Our work will serve to guide further advances in modeling these systems.