Here's a nice example that recently came up in an MSE question. Let $k$ be a field and let $Vect$ be the category of $k$-vector spaces and $Aff$ be the category of $k$-affine spaces. Every vector space is an affine space, giving a forgetful functor $F:Vect\to Aff$. On the other hand, every affine space has an associated vector space of the same dimension (the vector space of formal differences), giving a functor $G:Aff\to Vect$. The composition $GF:Vect\to Vect$ is naturally isomorphic to the identity. The composition $FG:Aff\to Aff$, on the other hand, is only unnaturally isomorphic to the identity: it takes every affine space to another affine space of the same dimension, but this cannot be made compatible with morphisms.

Here's a nice example that recently came up in an MSE question. Let $k$ be a field and let $Vect$ be the category of $k$-vector spaces and $Aff$ be the category of $k$-affine spaces. Every vector space is an affine space, giving a forgetful functor $F:Vect\to Aff$. On the other hand, every affine space has an associated vector space of the same dimension (the vector space of formal differences), giving a functor $G:Aff\to Vect$. The composition $GF:Vect\to Vect$ is naturally isomorphic to the identity. The composition $FG:Aff\to Aff$, on the other hand, is only unnaturally isomorphic to the identity: it takes every affine space to another affine space of the same dimension, but this cannot be made compatible with morphisms.