There is a spectral sequence $E_2^{i,j} := H^i(X,\underline{Ext}^j(\mathcal E,\mathcal F)) \implies Ext^{i+j}(\mathcal E,\mathcal F),$ for two sheaves $\mathcal E$ and $\mathcal F.$ (Here $\underline{Ext}$ denotes sheaf Ext.) Looking at the the low degree implications of this, one finds an exact sequence $0 \to H^1(X, \underline{Hom}(\mathcal E,\mathcal F)) \to Ext^1(\mathcal E,\mathcal F) \to H^0(X,\underline{Ext}^1(\mathcal E,\mathcal F)) \to H^2(X,\underline{Hom}(\mathcal E,\mathcal F)) \to \cdots.$ Taking $\mathcal E = \mathcal F$, the first map is the one you are asking about. If $\mathcal F$ is locally free than $\underline{Ext}^1$ vanishes, and one gets the isomorphism that you recalled. In general, this $\underline{Ext}^1$ need not vanish, the corresponding map in the exact sequence also need not vanish, and so there will be an injection $H^1(X,\underline{End}(\mathcal F)) \hookrightarrow Ext^1(\mathcal F)$ which is not surjective.
[Added: A typical example would come from taking $\mathcal F$ to be a skyscraper at a point, with $X$ positive dimensional.]