What is the formal difference between the light cone and a black hole?
In a general spacetime the distinction between the insides of a black hole and “insides” of a light cone (an absolute future of some event) could be quite subtle and even subjective. For example, among cosmological spacetimes with the Big Crunch there is a continuum of solutions smoothly interpolating from a black hole cosmology (where the “matter” consists of many black holes) to a uniform FLRW cosmology. The intermediate solutions could be subjectively interpreted as either growing black holes coalescing together or the cosmological density fluctuations slowing the expansion of lightcones.
So a formal definition of black hole exists only for certain classes of spacetimes with “nice” asymptotic behavior, such as asymptotically flat spacetimes, to which we restrict our further attention.
The key here is the future null infinity (denoted by $\mathscr{I}^+$). This is an important concept, and one should look for details in a book such as [1, 2], but informally $\mathscr{I}^+$ consists of all future endpoints of null geodesics “escaping to infinity”.
For spacetime points inside a black hole no causal trajectory (such as null geodesic) could reach $\mathscr{I}^+$, while for the points “inside” a normal light cone there are null geodesics reaching this null infinity. So the definition of a black hole region in [1 ] is: $B= M - J^-(\mathscr{I}^+)$, or “all the points of a manifold $M$ that do not lie in the past of $\mathscr{I}^+$” (in other words, all the events from which no signal could ever escape to infinity). In contrast, for an asymptotically flat spacetime without an event horizon all points of spacetime are in the past of $\mathscr{I}^+$, in other words $M=J^-(\mathscr{I}^+)$.
Note, that the existence of singularities inside of a black hole is not a necessity, but an artefact of an “ordinary” general relativity with a particular matter content. One could consider modified theories of relativity which do not have singularities but have black holes in the sense outlined above.
R.M. Wald, General Relativity, University of Chicago Press (Chicago, 1984).
S.W. Hawking and G.F.R. Ellis, The Large Scale Structure of Spacetime, Cambridge University Press (Cambridge, 1973).