Sol,
The effect you saw was likely one of your treadmill having a "breakpoint" in its algorithm which calculates calories. Simply put, it assumes you are walking at speeds up to (typically) 4.5 mph and running at speeds above that. Also, with the "net calorie burn" issue, they are backing out basal metabolic calorie burn, so since that number is the same, but the walking number is likely roughly twice the running number, the general rule that moving a mile is the same calories is likely closer to the truth than Amby's article leads us to believe, though, many of those calories are not from exercise.
Just to go through the exercise, if we consider 1200 calories to be daily basal metabolic rate (chosen for easy math, probably truer for a small woman, than someone bigger) - then the basal caloric contribution would be, and we use 3 mph as walking pace and 6 mph for running:
1200 calories/day x 1 day/24 hours x 1 hour/3 miles = 16 calories
while the running number would be half that, or 8 calories.
The gap closes a bit, but not vastly. I think most people who cited the "roughly the same calories per mile" did so based on knowing that running is less efficient than walking (except at very high speeds), and that it was an approximation. Looking at the total numbers, we'd wind up with walking burning 85+16 calories (101) and running burning 120+8 calories (128). Similar order of magnitude, though not the same. Is 100 about the same as 125? Most would say no, but when the need is to convince someone out of shape that walking may be more appropriate for them at a given stage, it may be a valid thing to play a bit loose with the truth, especially if it keeps them thinking long term in terms of health and weight loss.