What's the difference between the automorphism and isomorphism of graph?

By definition, an automorphism is an isomorphism from $G$ to $G$, while an isomorphism can have different target and domain.

In general (in any category), an automorphism is defined as an isomorphism $f:G \to G$.

As an example, consider the graphs $G$ and $G'$ on 4 vertices, labelled 1, 2, 3 and 4, where $G$ has edge set $\{\{1,2\},\{1,3\},\{2,3\},\{3,4\}\}$ and $G'$ has edge set $\{\{1,4\},\{2,3\},\{2,4\},\{3,4\}\}$. Sketch both of these graphs !

Then the permutation $\alpha = (1, 2, 3, 4)$ (in disjoint cycle notation) is an isomorphism from $G$ to $G'$. Why ? Because, applying $\alpha$ to the vertex labels in the edge set of $G$ we obtain the edge set of $G'$. Check this ! Since an isomorphism from $G$ to $G'$ exists, $G$ and $G'$ are isomorphic. However, since their edge sets are different, $G$ and $G'$ are not equal.

The permutation $\beta = (1, 2)$ is an isomorphism from $G$ to $G$ itself, that is, an automorphism of $G$, also known as a symmetry of $G$. Check this, as above !

An automorphism of a graph $\Gamma$ is an isomorphism from $\Gamma$ to itself.