We present an innovating sensitivity analysis for stochastic differential equations: We study the sensitivity, when the Hurst parameter $H$ of the driving fractional Brownian motion tends to the pure Brownian value, of probability distributions of smooth functionals of the trajectories of the solutions $\{X^H_t\}_{t\in \mathbb{R}_+}$ and of the Laplace transform of the first passage time of $X_H$ at a given threshold. We also present an improvement of already known Gaussian estimates on the density of $X^H_t$ to estimates with constants which are uniform w.r.t. $t$ in the whole half-line $\mathbb{R}_+ \setminus \{0\}$ and w.r.t. $H$ when $H$ tends to $\frac{1}{2}$.