I believe it depends on where the stores are located, not only state and town, but also district in a large city. In my town, there is very little difference in price between HD and Lowes when comparing identical products and neither HD nor Lowes is consistently lower. The same is true when comparing equivalent, but not identical, products. I do think that Lowes tends to carry higher priced brands and am not saying that those higher priced brands are better.
The same is true with food stores. For example Albertson's (Boise is the home of Albertson's) is a high priced store in Boise, but we don't have Safeway as a competitor. In Spokane, WA Albertson's prices are for more competitive as they have several chains as competitors.