For trying to understand how general a certain theorem is, I'm looking for an example of an essentially small abelian category which has enough projectives and enough injectives, but whose category of projectives is not equivalent to the category of injectives.

By a result of Auslander, each such category can be written as the category $\operatorname{mod} \operatorname{proj} \mathcal{A}$ of coherent/finitely presented functors (likewise as $\operatorname{mod}\operatorname{inj}\mathcal{A}$). Typical examples up to this point include the category of finite dimensional modules over a finite dimensional algebra, or generalisations of this, e.g. what is sometimes called a dualising $k$-variety, where the category is a $k$-category for a ground field and the $k$-duality provides an equivalence.