In category theory there is a general and very simple notion of subobject: a subobject of $X$ is a class of monomorphisms $A\to X$, where we identify $A\to X$ and $B\to X$ when they are related by an isomorphism $A\to B$. This is quite easy to use and understand, and works very well in a lot of categories.
But this is not what we do when we define subschemes: instead of using all monomorphisms, we use a somewhat ad hoc subclass of monomorphisms, namely immersions. Immersions are defined using two base cases, open immersions and closed immersions, which have a quite different flavor and behavior (not at all like in topology where open/closed subspaces can be easily defined in terms of each other).
There are historical reasons for this choice, and, of course, it works, but it has always bugged me on some level. Could someone give a clear and convincing mathematical argument for defining subschemes with immersions and not general monomorphisms? What I am looking for is something that would become true with this more general definition and makes you think "Oh yeah, we clearly don't want that to be true for what we call a subscheme." (Or of course equivalently something that becomes false and makes you think "We clearly wanted that to be true for subschemes".)
For instance, things like "a subscheme would not always be locally closed on a topological level" would not do it for me, since... so what? Subspaces in topology are not locally closed in general and no one is complaining.