Disclaimer: I am not an expert.

Curiously I just returned from a wonderful geometry seminar by Richard Thomas, who gave precisely a heuristic which addresses, perhaps if not your exact question, why you sometimes have to go to the derived category in order to get the equivalence.

This is in the context of homological mirror symmetry à la Kontsevich. Roughly speaking (did I mention I was not an expert?) homological mirror symmetry is a conjectural equivalence between derived categories: the derived category of coherent sheaves in a Calabi-Yau manifold and the derived Fukaya category of the mirror. The latter makes sense for symplectic manifolds, whereas the former for complex manifolds. Hence in a sense homological mirror symmetry is relating symplectic geometry to complex geometry.

Now symplectic geometry is very "floppy": symplectomorphisms are a dime a dozen and hence the Fukaya category has many autoequivalences, unlike the category of coherent sheaves due to the rigidity of complex geometry. Therefore one would not expect an equivalence of categories. What passing to the derived category does is to provide the extra autoequivalences which mirror symmetry requires.

I'm sure others here can make this much more precise and I, for one, would enjoy reading their version. This is why I'm making this answer into community wiki.