We consider a $d$-dimensional stochastic differential equation (SDE) of the form $d U_t = b(U_t) dt + σ\,d Z_t$, let $X_t$ be the solution if the driving noise $Z_t$ is a $d$-dimensional rotationally symmetric $α$-stable process ($1<α<2$), and let $Y_t$ be the solution if the driving noise is a $d$-dimensional Brownian motion. Continuing the work of [Deng,Schilling, Xu, Bernoulli, 23], we derive an estimate of the total variation distance $\|{\rm L} (X_{t})-{\rm law}(Y_{t})\|_{\rm TV}$ for all $t>0$, and we show that the ergodic measures $μ_α$ and $μ_2$ of $X_t$ and $Y_t$, respectively, satisfy $$\|μ_α-μ_2\|_{\rm TV} \leq \frac{Cd\log(1+d)}{α-1}(2-α).$$ We shall show that this bound is optimal with respect to $α$ by an Ornstein--Uhlenbeck SDE. Combining this bound with a recent interpolation result from \cite{HRW23}, we can derive a bound in Wasserstein-$p$ distance ($0< p <1$): \begin{gather*} \|μ_α-μ_2\|_{W_p} \leq\frac{Cd^{(p+3)/2}\log(1+d)}{α-1} (2-α). \end{gather*} {\bf Key Words:} Total variation distance, Wasserstein-$p$ distance, stochastic differential equation, Poisson equation, stable process.