I don't know exactly which sources and definitions you are looking at, but usually the distinction is the following:
An Internal Hom $[X,-]$ is the right adjoint to the functor $- \otimes X$ given by a tensor product.
An Exponential Object is the specific case of an internal hom where the tensor product is given by the cartesian product $- \times X$.
There are a good number of interesting tensor products that aren't the cartesian product, so we get many examples of internal homs which aren't exponential objects.
One good example comes from the category of vector spaces. We want the internal hom $[V,W]$ to be the space of linear maps from $V$ to $W$. To see what this should be the right adjoint of, let's consider linear maps $U \rightarrow [V,W]$.
If this internal hom was going to be an exponential object, we'd hope that these would correspond to linear maps $U \times V \rightarrow W$. However, this is not the case. Instead, each linear map $U \rightarrow [V,W]$ corresponds to a bilinear map $U \times V \rightarrow W$. So it's not an exponential object according to the definition.
How is $[V,W]$ an internal hom according to the definition? The answer to this gives us our first interesting example of a tensor product that isn't cartesian: the tensor product of vector spaces $U \otimes V$. There's a number of ways of defining it (for finite dimensional spaces, it has dimension $dim(U \otimes V) = dim(U) \times dim (V)$) but the key point is that linear maps $U \otimes V \rightarrow W$ correspond to bilinear maps $U \times V \rightarrow W$. Which, as we've already seen, correspond to linear maps $U \rightarrow [V,W]$.
In other words, the space of linear maps $V \rightarrow W$ is an internal hom, but the tensor product it corresponds to is the vector space tensor product, not the cartesian one.