Hermite-Gauss beams (HGBs), which have been presented as exact solutions to the paraxial wave equation in Cartesian coordinates, are natural resonating modes in stable laser resonators. In this work, we demonstrate that the apodized Hermite-Gauss beam (ApHGB), by a decaying exponential function exp(γx) (γ is the decaying or apodization parameter), is equivalent, under certain circumstances, to the apodized Airy beam (ApAiB). Based on the asymptotic treatment of the Hermite polynomials, we provide analytical expressions to describe their propagation dynamics in free space, then we investigate their similarity as a function of the apodization parameter. Moreover, by combining symmetric ApHGBs, we build dual and quad ApHGBs. The obtained numerical simulations confirm our analytical predictions. We believe that the ApHGB could be used as an alternative to the Airy beam in many applications.