Superconductivity in FeSe emerges from a nematic phase that breaks four-fold rotational symmetry in the iron plane. This phase may arise from orbital ordering, spin fluctuations or hidden magnetic quadrupolar order. Here we use inelastic neutron scattering on a mosaic of single crystals of FeSe, detwinned by mounting on a BaFe2As2 substrate to demonstrate that spin excitations are most intense at the antiferromagnetic wave vectors Q(AF) = (+/- 1, 0) at low energies E = 6-11 meV in the normal state. This two-fold (C-2) anisotropy is reduced at lower energies, 3-5 meV, indicating a gapped four-fold (C-4) mode. In the superconducting state, however, the strong nematic anisotropy is again reflected in the spin resonance (E = 3.6 meV) at Q(AF) with incommensurate scattering around 5-6 meV. Our results highlight the extreme electronic anisotropy of the nematic phase of FeSe and are consistent with a highly anisotropic superconducting gap driven by spin fluctuations.