We prove concentration inequalities and associated PAC bounds for continuous- and discrete-time additive functionals for possibly unbounded functions of multivariate, nonreversible diffusion processes. Our analysis relies on an approach via the Poisson equation allowing us to consider a very broad class of subexponentially ergodic processes. These results add to existing concentration inequalities for additive functionals of diffusion processes which have so far been only available for either bounded functions or for unbounded functions of processes from a significantly smaller class. We demonstrate the usefulness of the results by applying them in the context of high-dimensional drift estimation and Langevin MCMC for moderately heavy-tailed target densities. This is joint work with Cathrine Aeckerle-Willems and Claudia Strauch.