All models are wrong. Some are useful

In one, you mean? They get you perfectly fine answers around one

The theories on why are a fair bit beyond my knowledge of physics, but I do know that they're not necessarily the same kind of singularity. Inside a black hole (assuming our models are correct), spacetime curvature goes towards infinity. At the big bang, there may not have even been spacetime as we see it in our current universe, or whatever causes the expansion of spacetime may have been so powerful that it caused the earliest spacetime to not curve despite all the gravity

Different things.

The singularity of a black hole is located in space.

The initial singularity of the big bag happened "everywhere" the whole universe was supposed to have infinite density.

The mass of the black hole is finite. It's very dense but it have a quantifiable amount of mass.

For the big bang the mass was also infinite as far as we know. Everything was singularity, every "energy" in your body was part of that infinitely large singularity. Not only everything but everywhere. Where you sit there was singularity during the big bang. As far as we know every single point in space was part of the initial singularity. We don't come from a single point that exploded towards empty space. Expansion is more like the surface of a balloon. Maybe it's better to think of it as stretching rather than expanding.

Beyond that we don't know much about both, there are barriers which prevent direct observation of both.

The expansion of the universe is a completely different matter, as it's not only expanding, it's expanding faster that out gravitational models predict, like the universe is not only "ignoring" black holes, it's expanding despite all observable matter, and all untraceable matter (dark matter), and it's expanding faster and faster driven by an unknown phenomenon we call "dark energy" for giving it a name, because we have remotely not idea of what's going on.