Ferromagnets are known to support spin-polarized currents that control various spin-dependent transport phenomena useful for spintronics. On the contrary, fully compensated antiferromagnets are expected to support only globally spin-neutral currents. Here, we demonstrate that these globally spin-neutral currents can represent the Néel spin currents, i.e., staggered spin currents flowing through different magnetic sublattices. The Néel spin currents emerge in antiferromagnets with strong intrasublattice coupling (hopping) and drive the spin-dependent transport phenomena such as tunneling magnetoresistance (TMR) and spin-transfer torque (STT) in antiferromagnetic tunnel junctions (AFMTJs). Using RuO_{2} and Fe_{4}GeTe_{2} as representative antiferromagnets, we predict that the Néel spin currents with a strong staggered spin polarization produce a sizable fieldlike STT capable of the deterministic switching of the Néel vector in the associated AFMTJs. Our work uncovers the previously unexplored potential of fully compensated antiferromagnets and paves a new route to realize the efficient writing and reading of information for antiferromagnetic spintronics.