In the last decades a large effort has been devoted to the study of water confined in hydrophobic geometries at the nanoscale (tubes, slit pores), because of the multiple technological applications of such systems, ranging from drugs delivery to water desalinization devices. To our knowledge, neither numerical/theoretical nor experimental approaches have so far reached a consensual un-derstanding of structural and transport properties of water under these conditions. In this work, we present molecular dynamics simulations of TIP4P/2005 water under different hydrophobic nanoconfinements (slit pores or nanotubes, with two degrees of hydrophobicity) within a wide temperature range. On the one side, water is more structured near the hydrophobic walls, inde-pendently on the confining geometries. On the other side, we show that the combined effect of confinement and curvature leads to an enhanced diffusion coefficient of water in hydrophobic nan-otubes. Finally, we propose a confined Stokes-Einstein relation to extract viscosity from diffusivity, whose result strongly differs from the Green-Kubo expression that has been used in previous work. We discuss the shortcomings of both approaches, which could explain this discrepancy.