This Letter presents the measurement of the energy-dependent neutrino-nucleon cross section in tungsten and the differential flux of muon neutrinos and antineutrinos. The analysis is performed using proton-proton collision data at a center-of-mass energy of 13.6 TeV and corresponding to an integrated luminosity of (65.6±1.4)  fb-1. Using the active electronic components of the FASER detector, 338.1±21.0 charged current muon neutrino interaction events are identified, with backgrounds from other processes subtracted. We unfold the neutrino events into a fiducial volume corresponding to the sensitive regions of the FASER detector and interpret the results in two ways: (i) we use the expected neutrino flux to measure the cross section, and (ii) we use the predicted cross section to measure the neutrino flux. Both results are presented in six bins of neutrino energy, achieving the first differential measurement in the TeV range. The observed distributions align with standard model predictions. Using this differential data, we extract the contributions of neutrinos from pion and kaon decays.