"Natural" categories where monomorphisms differ from injective morphisms

In category theory there is no notion of injectivity for morphisms; that's why we have monomorphisms instead. What you call injectivity is sometimes called being a split monomorphism (see e.g. this blog post), and in general it is much stronger than being a monomorphism. As a simple example, in the category of fields, every morphism is a monomorphism, but the split monomorphisms are precisely the isomorphisms.

For a large class of examples, in an abelian category $C$, if $f : A \to B$ is a monomorphism then it fits into a short exact sequence

$$0 \to A \to B \to B/A \to 0$$

and $f$ is a split monomorphism if and only if this short exact sequence splits. The condition that every short exact sequence splits, or equivalently that every monomorphism is a split monomorphism, is a strong condition on $C$ called semisimplicity that is rarely satisfied in practice: for example, if $C = \text{Mod}(R)$ is the category of modules over a ring $R$, then it holds if and only if $R$ is semisimple as a ring.

It's more typical to define injectivity with respect to a forgetful functor $F : C \to \text{Set}$, as follows.

Definition: A morphism $f$ is injective, or maybe more precisely $F$-injective, if $F(f)$ is injective.

This condition is sometimes called "injective on underlying sets" for clarity. In this setting we have the following results.

Proposition: If $F$ is faithful, then every injective morphism is a monomorphism.

This condition is usually part of the definition of a forgetful functor, or more precisely part of the definition of a concrete category, so it's typically automatic.

Proposition: If $F$ preserves pullbacks, then every monomorphism is injective.

This condition is frequently satisfied in practice; $F$ is frequently representable, in which case it preserves all limits, not just pullbacks.

This is why injective morphisms and monomorphisms typically coincide in familiar examples of concrete categories. (The situation is very different for surjective morphisms and epimorphisms.) So, at this point, we might ask:

Is there a "natural" example of a concrete category in which not every monomorphism is injective in this sense?

It's a bit tricky to find such examples because nearly every forgetful functor that comes up in practice preserves pullbacks. But here is one: consider the category of pointed, path connected, locally path connected topological spaces, with the obvious forgetful functor passing through topological spaces. Every covering map is a monomorphism in this category; this is a restatement of one of the lifting properties of covering maps. But most covering maps are not injective.