Here is an example: Let $A \to B \to C \to 0$ be a sequence in $\mathcal{C}$. Then it becomes exact in $\widehat{\mathcal{C}^{\mathrm{op}}}^{\mathrm{op}}$ (where $\widehat{\mathcal{C}} := [\mathcal{C}^{\mathrm{op}},\mathsf{Ab}]$) after applying $\mathcal{C} \hookrightarrow \widehat{\mathcal{C}^{\mathrm{op}}}^{\mathrm{op}}, X \mapsto \hom(X,-)$ if and only if $B \to C$ is a cokernel of $A \to B$ (immediate). On the other hand, it becomes exact in $\widehat{\mathcal{C}}$ after applying $\mathcal{C} \hookrightarrow \widehat{\mathcal{C}}, X \mapsto \hom(-,X)$ if and only if it is isomorphic to a sequence of the form $F \oplus E \xrightarrow{p} C \oplus E \xrightarrow{q} C \to 0$, where $E,F,C \in \mathcal{C}$, $q$ is the projection and $p$ is the projection $F \oplus E \to E$ followed by the inclusion $E \to C \oplus E$ (exercise). Thus we get a split exact complex. Therefore, exactness in $\widehat{\mathcal{C}}$ applies the one in $\widehat{\mathcal{C}^{\mathrm{op}}}^{\mathrm{op}}$, but not vice versa.

Similarily, $B \to C \to 0$ becomes exact in $\widehat{\mathcal{C}^{\mathrm{op}}}^{\mathrm{op}}$ if and only if $B \to C$ is an epimorphism, and it becomes exact in $\widehat{\mathcal{C}}$ if and only if $B \to C$ is a split epimorphism.

By duality, $0 \to A \to B \to C$ becomes exact in $\widehat{\mathcal{C}}$ if and only if $A \to B$ is a kernel of $B \to C$, and it becomes exact in $\widehat{\mathcal{C}^{\mathrm{op}}}^{\mathrm{op}}$ if and only if it is a split exact complex.

Combining these results, we see that $0 \to A \to B \to C \to 0$ becomes exact in $\widehat{\mathcal{C}^{\mathrm{op}}}^{\mathrm{op}}$ if and only if it becomes exact in $\widehat{\mathcal{C}}$ if and only if it is a split exact complex.

Even for abelian $\mathcal{C}$, this is not the "correct" notion of exactness. I doubt that there is anything interesting which can be said about sequences of the form $A \to B \to C$.

Edit: For abelian $\mathcal{C}$ the sequence $A \to B \to C$ becomes exact in $\widehat{\mathcal{C}^{\mathrm{op}}}^{\mathrm{op}}$ iff $A \to B \to C$ is exact and the image of $B \to C$ is a direct summand of $C$. See mathunderflow 360234.