These are two different volume integrals, see below. To see they are different, you could for example take $\textbf{F}(\textbf{r})=\textbf{r}$. Then the curl of $\textbf{F}$ vanishes, so the second integral is zero, but the first integral is nonzero for arbitrary $V$.

$$\textbf{curl}\biggl(\oint_{S}^{}{\textbf{n}\times\dfrac{\textbf{F}(\textbf{r}^\prime)}{|\textbf{r}-\textbf{r}^\prime|} ~ds}\biggr) = \int_{V}^{}\frac{\partial}{\partial\textbf{r}}\times\biggl({\frac{\partial}{\partial\textbf{r}'}\times \dfrac{ \textbf{F} (\textbf{r}^\prime)}{|\textbf{r}-\textbf{r}^\prime|}\biggr)~d^3\textbf{r}'}$$

$$\oint_{S}^{}{\textbf{n}\times \dfrac{ \textbf{curl} (\textbf{F}) (\textbf{r}^\prime)}{|\textbf{r}-\textbf{r}^\prime|}~ds}=\int_{V}^{}{\frac{\partial}{\partial\textbf{r}'}\times \biggl(\frac{1}{|\textbf{r}-\textbf{r}'|}\frac{\partial}{\partial\textbf{r}'}×\textbf{F}(\textbf{r}')\biggr)~d^3\textbf{r}'}$$

These are two different integrals. To see they are different, you could for example take $\textbf{F}(\textbf{r})=\textbf{r}$. Then the curl of $\textbf{F}$ vanishes, so the integral on the right-hand-side is zero, but the left-hand-side integral does not vanish for arbitrary $V$ (without any spherical symmetry).