In the case of differential geometry everything reduces to vector spaces. Let $x \in X$. Then at any point $(x, y) \in X \times X$ $$ T_{(x, y)} X \times X = T_x X \oplus T_y X . $$ Using this identification the tangent to the diagonal at a point $(x, x)$ is the subspace of $T_{(x, x)} X \times X$ given by $$ T_{(x, x)} (\Delta) = \lbrace (\xi, \xi ) \mid \xi \in T_x X \rbrace \subset T_{(x, x)} X \times X. $$ On the other hand the normal is the quotient $$ (T_{(x, x)} X \times X) / T_{(x, x)} \Delta $$ and we can identify this with $T_x X$ in at least two slightly different ways. Either $$ \iota_1 \colon \xi \mapsto (\xi , - \xi) + T_{(x, x)} \Delta $$ or $$ \iota_2 \colon \xi \mapsto (-\xi , \xi) + T_{(x, x)} \Delta . $$
We can of course also identify $T_x X$ and $T_{(x, x)} \Delta $ by $ \xi \mapsto (\xi, \xi)$.
I guess one explanation for the two identifications of the normal bundle is that there is involution $\tau \colon X \times X \to X \times X $ given by $\tau(x, y) = (y, x)$ which fixes the diagonal pointwise and hence acts trivially on the tangent space to the diagonal. As a result it descends to an action on the normal bundle which interchanges the two identifications $\iota_1$ and $\iota_2$, that is $\tau \circ \iota_1 = \iota_2$
