I am looking for a nontrivial category such that its regular monics are not the same as its split monics.
The necessary definitions are found here.
A necessary requirement is that our category is not a (Epi, RegMonic)-factorization system, ie each arrow has an orthogonal epic/regular monic factorization.
Moreover, I am aware of trivial cases such as the category consisting of 3 objects and 4 non-identity arrows, $\{f\colon X\to Y,g\colon Y\to Z,h\colon Y\to Z,k=gf=hf\colon X\to Z\}$.
In $\mathbf{Set}$ the unique function:
$\emptyset \rightarrow \{\emptyset\}$
is a regualr monomorphism, which is not split. More generally, in any elementary topos for every object $X$ the unique (necessary regular mono) morphism:
$$0 \rightarrow X$$
is non-split unless $X = 0$.