I'm not really sure what the cost/mile is for heavy, but the civil construction cost is ~$25-$30M per mile for lightrail. Add another $10-$15M per mile for the electrical/communication system installation, configuration, and testing. I should also qualify this cost that it is a "turn-key" cost which includes stations, any roadway relocations, property acquisition, parking lots, everything. I have no idea what the cost/mile is for only the rail.
Because of the weight difference, the rail/ballast specs are significantly cheaper. Rails are smaller, ballast section is thinner, but the subgrade prep is about the same. Because we have to maintain a smooth ride, the rail can't deflect the same as a heavy rail can, plus lightrail is welded in place as one continuous piece to eliminate the clink-clink commonly associated with heavy (plus the rail serves as the "ground" for the electric propulsion).