Thanks to Daniel and Rohit for the tips ... which also led to the discovery of the WeightedAdjacencyGraph[] constructor. Here's what I discovered from some tests with my data ... all times local on a 5 yr-old Windows 10 box.
125 vertices, 7750 undirected weighted edges.
Unweighted graph, Graph[edges] ? 0.006 sec.
Weighted graph by list of weights Graph[vertices, edges, EdgeWeight -> edgeWeightsList] ? 0.006 sec.
Weighted graph by WeightedAdjacencyMatrix[vertices,weightedAdj] ? 0.05 sec.
Weighted graph by rules (e -> w) Graph[vertices, edges, EdgeWeight -> edgeWeightsRules] ? 1.8 sec.
Weighted graph by Annotation, Graph[annotatedEdges] ? 3.5 hours.
All the weighted graphs above have equivalent WeightedAdjacencyMatrix's, but the specialized constructor produces a graph containing self-edges that are infinitely weighted, so its graph object is not equal to the others.
Here's my test code, excluding the last one which ran overnight in a separate notebook.
