Differential cross sections are measured for the standard model Higgs boson produced in association with vector bosons ($W$,$Z$) and decaying to a pair of$b$quarks. Measurements are performed within the framework of the simplified template cross sections. The analysis relies on the leptonic decays of the$W$and$Z$bosons, resulting in final states with 0, 1, or 2 electrons or muons. The Higgs boson candidates are either reconstructed from pairs of resolved$b$-tagged jets, or from single large-radius jets containing the particles arising from two$b$quarks. Proton-proton collision data at$\sqrt{s}=13\mathrm{TeV}$, collected by the CMS experiment in 2016–2018 and corresponding to a total integrated luminosity of$138{\mathrm{fb}}^{-1}$, are analyzed. The inclusive signal strength, defined as the product of the observed production cross section and branching fraction relative to the standard model expectation, combining all analysis categories, is found to be$\mu =1.15_{-0.20}^{+0.22}$. This corresponds to an observed (expected) significance of 6.3 (5.6) standard deviations.