I know that the Big Bang theory states that the early Universe had a very high temperature and was extremely dense (thus causing a gravitational/space-time collapse) but if that's the case then wouldn't it be greater than that of a black hole (and thus become a black hole)? If not, how does the Big Bang create something denser/hotter without becoming a black hole or violating conservation of energy?
In order for gravitational collapse to happen, there has to be a way for matter to pick a point toward which to collapse. It's true that the early universe was very dense, but it also comprised an infinite and uniform density distribution. This means that every point would be just as good a "center" as any other point. Since no point can be the center of collapse, you get no collapse.
The Schwarzschild solution for a black hole assumes the local spacetime is both largely flat (largely in universe terms) and static in time.
In comparison, the early universe was rapidly expanding and highly "bent". Simply put, the Schwarzschild solution does not apply.
A key difference is that the universe has a singularity in the past, while a black hole's singularity is in the future light cone. In that respect, the universe is sort of like a time-reversed black hole, where everything is "falling out" rather than in. But it's not really like that either, there's all sorts of subtle differences.
Just found this, he says it better than I ever could.