Really I think this question boils down to what the physical significance of a connection is.
Physically, we can think of a symplectic manifold $(\mathcal{M},\omega)$ as essentially a phase space.
To (pre-)quantize the manifold we must choose a connection over a vector bundle $E \xrightarrow{\pi}\mathcal{M}$ (not worrying about principle bundles/gauges).
Given any bundle, there's an affine spaces of connections associated to it.
Physically speaking, I suppose we can see a choice of connection as a means of describing the evolution of the system - since a connection [assuming we are on a 'nice' manifold] provides a unique geodesic curve $\gamma_{p}$ through each point $p \in \mathcal{M}$, i.e choice of connection means we produce a flow, $\sigma : \; \mathbb{R}\times \mathcal{M} \rightarrow \mathcal{M} \; \big|\; \sigma_{p}(t):=\gamma_{p}(t)$.
However,choosing a connection isn't the only way of generating a flow on a manifold. For example,if the manifold is also a lie group, then we can look at the left invariant vector fields.
- Why then is there so much interest in quantizing the space of connections associated to a given symplectic manifold? [i.e what is special about the 'evolutions' determined by these]
Things get even more confusing to me if considering principal bundles -since I'm still unsure of how the Ehrsmann connections is related to the usual idea of a connection on a vector bundle...