Consider the open embedding of schemes $\mathbf A^2\setminus\{(0,0)\}\subset \mathbf A^2$ (say over $\mathbb C$). This induces a restriction functor on the stacks $$\mathrm{Vect}_n(\mathbf A^2)\to \mathrm{Vect}_n(\mathbf A^2\setminus \{(0,0)\}).$$ Here $\mathrm{Vect}_n(X)$ is the functor $S\mapsto \{\mbox{vector bundles on }X\times S,\mbox{ with isomorphisms between them}\}$.
The restriction functor is not an equivalence. However, Hartogs' theorem tells us that it is an equivalence on $\mathbb C$-rational points.
Now the question is: are there any "topological" invariants that tell us that the derived categories of sheaves of abelian groups of these two stacks are non-isomorphic?
NOTE: I use the tag "algebraic stacks" although these stacks are not Artin stacks.