Is there any generalization of Riemann Mapping theorem?

Question

Given any two regions in complex plane when can we say they are conformally equivalent? I mean does there exists some "complex-geometric" invariant which determines whether two regions are conformally equivalent or not. Obviously topological invariant won't work as both are at least diffeomorphic. Can you please provide some intuition why $B(0,1) \setminus \{0\}$ is not conformally equivalent with any annulus with some positive radius? (I am not asking for proof). I understand some angle "should" be distorted because any angle preserving and orientation preserving map is holomorphic. Orientation preserving still understandable but I am not getting where is angle distortion happening? Thank you.