Computational hardness for sampling a uniform matching
Concerning your general question, I think the paper by Aaronson and Arkhipov [1] on boson sampling is a good example. They show that if there exists a classical poly-time algorithm that exactly (or at least "extremely accurate" as they state it) samples from a certain distribution, then the polynomial hierarchy collapses up to some level.

To prove the same thing for an approximate sampling algorithm, they must rely on some unproven conjectures.

[1] Aaronson, Scott, and Alex Arkhipov. "The computational complexity of linear optics." Proceedings of the forty-third annual ACM symposium on Theory of computing. ACM, 2011.