Let $F,G$ be quasicoherent sheaves of $\mathcal{O}_X$modules on a scheme $X$.
What is exactly the difference between the derived functors $\mathcal{Ext}^i(F,G)$ and $Ext^i_X(F,G)$?
By definition $Ext^i_X(F,G)$ arises from the the global section functor $U \to \Gamma(U, F^{\vee} \otimes G)$ while $\mathcal{Ext}^i(F,G)$ from the Hom functor $U \to \mathcal{Hom}_U(F|_U,G|_U)$.
Both are calculated using injective resolutions $I_{\bullet}$ of $F$ or $G$.
Futhermore the functors are essentially the same since
(*) $$\Gamma(U, F^{\vee} \otimes G)= \Gamma(U, \mathcal{Hom}(F, \mathcal{O}_X) \otimes G)= \mathcal{Hom}(\mathcal{O}_X, \mathcal{Hom}(F, \mathcal{O}_X)\otimes G)= \mathcal{Hom}_X(\mathcal{F}, \mathcal{G})$$
So why generally $\mathcal{Ext}^i(F,G)$ and $Ext^i_X(F,G)$ should be distinguished?
Till now - according to line (*) - they arise from the same functor, or am I wrong?