In crystalline cohomology, you want to lift your variety in prime characteristic $p$ to a $p$-adic variety and then take its de Rham cohomology. But there's no natural lift. If you're Dwork, you just pick a lift and compute. But if you're Grothendieck, you look at the category of all lifts, and then things work by magic.
In fact, here the failure of uniqueness is more extreme, because you might not even have existence. In other words, your variety might not lift at all. But it does always lift locally, so you need to do everything for all sufficiently small open subvarieties. This is a common theme in geometry, where the failure of a local construction to be unique can prevent the global construction from existing at all.
I mentioned this example before here.