To add to the other good answers here, there's a family of examples that could be seen as a bit trivial. But in a sense they give the simplest answer to your question.

Let $X$ be a partially ordered set, viewed as a category in the usual way: the objects of the category are the elements of $X$, and $\textrm{Hom}(x, y)$ has either $1$ or $0$ elements according to whether $x \leq y$ or not. If $X$ is a meet-semilattice, i.e. any finite set of elements has a greatest lower bound, then the corresponding category has finite products. It's cartesian closed if for any two elements $y$ and $z$ there's an element $y \to z$ with the property that for all $x \in X$, $$ x \wedge y \leq z \iff x \leq y \to z. $$ Here $\wedge$ denotes greatest lower bound. So $y \to z$ is the internal hom $z^y$. A poset with this property is more or less what's called a Heyting algebra.

I think this is an enlightening family of examples because in a poset, the external homs are pretty trivial (sets with at most one element), whereas the internal homs can be informative. For example, in a power set they give you the notion of complement (exercise!).

You can extend this family a bit. Again take an ordered set $X$, regarded as a category in the usual way. But now think about non-cartesian monoidal structures on it. For instance, take $X = \mathbb{R}$ with its usual ordering. The cartesian monoidal structure is $\mathrm{min}$, but we could instead use $+$. Then the internal hom is given by $$ y \to z = z - y. $$ Is that "unexpected"? That depends on your intuition. But in this example the external hom only tells you the sign of $z - y$, whereas the internal hom tells you what it's actual value. In other words, it produces the operation of subtraction, which historically has proved quite important.

To add to the other good answers here, there's a family of examples that could be seen as a bit trivial. But in a sense they give the simplest answer to your question.

Let $X$ be a partially ordered set, viewed as a category in the usual way: the objects of the category are the elements of $X$, and $\textrm{Hom}(x, y)$ has either $1$ or $0$ elements according to whether $x \leq y$ or not. If $X$ is a meet-semilattice, i.e. any finite set of elements has a greatest lower bound, then the corresponding category has finite products. It's cartesian closed if for any two elements $y$ and $z$ there's an element $y \to z$ with the property that for all $x \in X$, $$ x \wedge y \leq z \iff x \leq y \to z. $$ Here $\wedge$ denotes greatest lower bound. So $y \to z$ is the internal hom $z^y$. A poset with this property is more or less what's called a Heyting algebra.

I think this is an enlightening family of examples because in a poset, the external homs are pretty trivial (sets with at most one element), whereas the internal homs can be informative. For example, in a power set they give you the notion of complement (exercise!).

You can extend this family a bit. Again take an ordered set $X$, regarded as a category in the usual way. But now think about non-cartesian monoidal structures on it. For instance, take $X = \mathbb{R}$ with its usual ordering. The cartesian monoidal structure is $\mathrm{min}$, but we could instead use $+$. Then the internal hom is given by $$ y \to z = z - y. $$ Is that "unexpected"? That depends on your intuition. But in this example the external hom only tells you the sign of $z - y$, whereas the internal hom tells you its actual value. In other words, it produces the operation of subtraction, which historically has proved quite important.