  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 $t\bar{t}H$ 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 ($p^{\text{H}}_\text{T}$), 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 $p^{\text{H}}_\text{T}>200$ GeV ($p^{\text{H}}_\text{T}<200$ GeV) is
${1.29}^{+0.39}_{-0.34}$ (${0.12}^{+0.34}_{-0.33}$). This is the first VBF
measurement for the higher-$p^{\text{H}}_\text{T}$ criteria, and the most
precise for the lower-$p^{\text{H}}_\text{T}$ 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
$c_{H\tilde{W}}$.
