We obtain non-asymptotic Gaussian concentration bounds for the difference between the invariant measure $ν$ of an ergodic Brownian diffusion process and the empirical distribution of an approximating scheme with decreasing time step along a suitable class of (smooth enough) test functions f such that f -- $ν$(f) is a coboundary of the infinitesimal generator. We show that these bounds can still be improved when the (squared) Fr{ö}benius norm of the diffusion coefficient lies in this class. We apply these bounds to design computable non-asymptotic confidence intervals for the approximating scheme. As a theoretical application, we finally derive non-asymptotic deviation bounds for the almost sure Central Limit Theorem.