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They are both geodesic singularities, meaning certain families of geodesics essentially "end" at them. However to put it simply the black hole singularity is a singularity in space and the big bang is a singularity in time: try to evolve any geodesic into a black hole and it will terminate within the black hole; try to evolve any geodesic backwards in time and it will terminate (or begin) at the big bang.
