We discuss the parallel between the third-order Moore-Gibson-Thompson equation partial derivative(ttt)u + alpha partial derivative(tt)u - beta Delta partial derivative(tu) - gamma Delta u = 0 depending on the parameters alpha, beta, gamma > 0, and the equation of linear viscoelasticity partial derivative(tt)u(t) - kappa(0)Delta u(t) - integral(infinity)(0) kappa '(s)Delta u(t-s) ds = 0 for the particular choice of the exponential kernel kappa(s) = ae(-bs) + c with a, b, c > 0. In particular, the latter model is shown to exhibit a preservation of regularity for a certain class of initial data, which is unexpected in presence of a general memory kernel kappa.