We take advantage of recent results on optimal quantization theory (see~\cite{GraLusPag1, PagWil}) to improve the quadratic optimal quantization error bounds for backward stochastic differential equations (BSDE in short) and nonlinear filtering problems. To achieve it, one of the main ideas used for both problems is the use the orthogonality property of the conditional expectation for the mean-quadratic norm. When permitting some involving functions to be less regular than what is usually needed, the analysis of the nonlinear filtering error bounds brings into play the so-called mismatch property, namely the fact that the quadratic optimal quantizers of size $N$ used to approximate $\mathds R^d$-valued random vectors in $L^2$ by a nearest neighbor projection (Voronoi quantization) at a $N^{-\frac 1d}$ still perform this approximation at the same rate in $L^s$, $2\le s\le 2+d$.