symmetry - Why do we classify states under covering groups instead of the group itself?

Why do we always classify states under covering group representations instead of the group itself? For example see the following picture I lifted from 'Symmetry in physics' by Gross

So in the first example, why not classify states under SO(3) instead of SU(2)?

EDIT: From a physicists perspective, I know enough that we consider projective representations of groups in quantum mechanics because those are the most general thing that keeps $ |\langle \phi|\psi\rangle|^2 $ invariant. But Gross above seems to indicate that representations of the covering group are a better way to go. Furthermore, covering groups are always simply connected, to where-as groups with projective representations are in general not simply connected - is this property of (universal) covering groups related to why we use them in these sorts of cases?