Imagine three nodes (e.g. cities), A, B, C.
A and B are connected by multiple paths. B and C have 1 path. A and C have one path. If the question is, given paths of equal length, which two adjacent cities are most important (population?), edge weighting is irrelevant.

Edge weighting isn't bad, but remember edges are defined by two nodes- not 1- which can differentiate a problem definition.