I would be tempted to rephrase my question as : why do people seem to care only about the curvature of a connection on fiber bundles ?
Indeed, the curvature gives the vertical part of the commutator of horizontal fields (the horizontal part is the lift of base commutator), while the commutator of vertical fields has nothing to do with the connection. So it remains, for me, aside from the curvature, to understand the commutator of a horizontal and a vertical field.
In the case of principal bundles and principal (=invariant) connections, I understand it well, since we can use the action of the group. But what happens then in general for fiber bundles? Doesn't it give information on how the horizontal distribution vary in one fiber ?