Why 'which store is cheaper' is the wrong question
Home Depot and Lowe's are close competitors, which is the whole point. They shadow each other on headline items and make their margin on the rest. So neither can be cheaper across the board, by design. If one were clearly cheaper on everything, the other would not survive.
What actually happens is per item. On any given week, one store has the better number on framing lumber, the other on sheathing, a third category swings back. Add in fasteners, hardware, concrete, insulation, and finishes, and the "cheapest store" for a full list is a patchwork, not a single logo.
Prices also move. A great price last month tells you nothing about this week, and it is not the same from one region to the next. So even if you did the homework once, it goes stale fast.