By definition: \begin{align}\tag{1} A_{\mu,\nu} &\equiv \frac{\partial}{\partial x^{\nu}} \, A_{\mu}, \\[1ex] \tag{2} A_{\mu;\nu} &\equiv \frac{\partial}{\partial x^{\nu}} \, A_{\mu} - \Gamma_{\nu \mu}^{\lambda} \, A_{\lambda},\end{align} If the connection is symetric: $\Gamma_{\mu \nu}^{\lambda} = \Gamma_{\nu \mu}^{\lambda}$ (Levi-Civita connection AKA Christoffel symbols), then the antisymetric expression $A_{\mu;\nu} - A_{\nu;\mu}$ gives the same as $A_{\mu,\nu} - A_{\nu,\mu}$.

By definition: \begin{align}\tag{1} A_{\mu,\nu} &\equiv \frac{\partial}{\partial x^{\nu}} \, A_{\mu} \equiv \partial_{\nu} \, A_{\mu}, \\[1ex] \tag{2} A_{\mu;\nu} &\equiv \frac{\partial}{\partial x^{\nu}} \, A_{\mu} - \Gamma_{\nu \mu}^{\lambda} \, A_{\lambda} \equiv \nabla_{\nu} \, A_{\mu},\end{align} If the connection is symetric: $\Gamma_{\mu \nu}^{\lambda} = \Gamma_{\nu \mu}^{\lambda}$ (Levi-Civita connection AKA Christoffel symbols, which means no torsion), then the antisymetric expression $A_{\mu;\nu} - A_{\nu;\mu}$ gives the same as $A_{\mu,\nu} - A_{\nu,\mu}$.