They are two very different physical phenomena described by two very different sets of equations in general relativity. The black hole 'solution' is a singularity of the gravitational field within a pre-existing, 4-d space-time. Black holes are embedded in space, and there is a coordinate transformation which can be used to take you from the interior of the black hole inside the event horizon, to the external space outside the horizon.

The 'Big Bang solution' has no external space in which it is embedded. The space defined by this solution is a dynamic one which is always in a state of change as the universe expands or collapses. It is like asking why apples are not bananas since they are both fruits.

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