Differential measurements of Higgs boson production in the tau-lepton-pair decay channel are presented in the gluon fusion, vector-boson fusion (VBF), VH and t (t) over barH associated production modes, with particular focus on the VBF production mode. The data used to perform the measurements correspond to 140 fb(-1) of proton-proton collisions collected by the ATLAS experiment at the LHC. Two methods are used to perform the measurements: the Simplified Template Cross-Section (STXS) approach and an Unfolded Fiducial Differential measurement considering only the VBF phase space. For the STXS measurement, events are categorized by their production mode and kinematic properties such as the Higgs boson's transverse momentum (p(T)(H)), the number of jets produced in association with the Higgs boson, or the invariant mass of the two leading jets (m(jj)). For the VBF production mode, the ratio of the measured cross-section to the Standard Model prediction for m(jj) > 1.5 TeV and p(T)(H) > 200 GeV (p(T)(H) < 200 GeV) is 1.29(-0.34)(+0.39) (0.12(-0.33)(+0.34)). This is the first VBF measurement for the higher-p(T)(H) criteria, and the most precise for the lower-p(T)(H) criteria. The fiducial cross-section measurements, which only consider the kinematic properties of the event, are performed as functions of variables characterizing the VBF topology, such as the signed Delta phi(jj) between the two leading jets. The measurements have a precision of 30%-50% and agree well with the Standard Model predictions. These results are interpreted in the SMEFT framework, and place the strongest constraints to date on the CP-odd Wilson coefficient c(H (W) over tilde).