The paper presents the results of studies of the electric and dielectric properties of dimethyl sulfoxide (DMSO) and acetonitrile (ACN), two strongly polar liquids composed of the molecules of the same dipole moment value (mu approximate to 4 D) but of a quite different static dielectric permittivity (epsilon(DMSO)(S) >> epsilon(ACN)(S)) It was shown that the activation energies for both the dc ionic conductivity (sigma(DC)) and the viscosity (eta) are two times higher for DMSO than for ACN; however, for both of the liquids, the temperature dependence of the product sigma(DC)eta is quite close to the prediction of the Stokes-Einstein relation. The dielectric results are interpreted in terms of the intermolecular dipole-dipole coupling. An exceptional behavior of DMSO most certainly results from its "monomolecularity", i.e., from the lack of the dipolar coupling in that strongly polar liquid. The effect is a consequence of a very specific structure of the DMSO molecule where its rotational dynamics makes the intermolecular dipole-dipole coupling very unfavorable, in contrast to the ACN molecules.