>The fact that both have the same name

They don't just have the same name, they are the same thing.

A Schwarzschild black hole has both: a removable singularity at the event horizon that is just an artefact of a particular choice of coordinates and a true non-removable mathematical singularity at r=0 where curvature really does go to infinity. It also wouldn't be much of an issue in classical physics, because this singularity is always hidden from outside observers, so the mathematical weirdness there can't screw with your normal predictions in space outside the black hole. The problems start once you consider quantum mechanics, because any such singularity will break unitarity (a fancy way of saying that probabilities must add up to 1), which means your theory as a whole can no longer make predictions. This has opened a whole can of worms with a bunch of solution attempts, which are all sadly untestable for the foreseeable future.