In this paper, we consider the spatially inhomogeneous diffusively driven inelastic Boltzmann equation in different cases: the restitution coefficient can be constant or can depend on the impact velocity (which is a more physically relevant case), including in particular the case of viscoelastic hard-spheres. In the weak thermalization regime, i.e when the diffusion parameter is sufficiently small, we prove existence of global solutions considering both the close-to-equilibrium and the weakly inhomogeneous regimes. We also study the long-time behavior of these solutions and prove a convergence to equilibrium with an exponential rate. The basis of the proof is the study of the linearized equation. We obtain a new result on it, we prove existence of a spectral gap in weighted (stretched exponential and polynomial) Sobolev spaces, more precisely, there is a one-dimensional eigenvalue which is negative and which tends to $0$ when the diffusion parameter tends to $0$. To do that, we take advantage of the recent paper \cite{GMM} where the spatially inhomogeneous equation for elastic hard spheres has been considered. As far as the case of a constant coefficient is concerned, the present paper improves similar results obtained in \cite{MM2} in a spatially homogeneous framework. Concerning the case of a non-constant coefficient, this kind of results is new and we use results on steady states of the linearized equation from \cite{AL3}.