Lowe's vs Home Depot? There is no one set answer.
I went through the same decision process earlier this year and ended up using both. I looking around, I have found that - in general - Lowe's tends to have higher quality lumber than HD. This is not only true in the two areas I shop, but matches comments online from people in other areas. To know for sure the quality in your area, it should be easy enough to visit both stores.
That said, lumber quality is not that great is either store. You can spend a while looking for straight framing lumber. For a large project it isn't feasible to cherry pick - the selection is usually that bad. Overall it is "workable" if you account for a little extra waste. If there is a lumberyard nearby, it may be worth taking a look. Talk to the folks there and see what they can do on price. Often, they will _try_ to compete with the big box stores. Sometimes, they can match or beat HD/Lowes on a larger order.
Price comparisons are easy. Both Lowe's and HD allow you to choose your store and shop online. You can even build an online shopping list/order for pick up in-store. View the items and see the pricing. Depending on the items and quantity, the price difference can be substantial. Most stores, the manager also has some discretion on pricing. On a large order, you can sometimes negotiate.
I had a large lumber order. HD ended up having the lower price, along with some specific sizes I needed. I made a "shopping list" online with the full material list and went into the store. For whatever reason, the store did not carry some of the items listed online (for that store). The manager "upgraded" me on those items (i.e. they didn't carry 2x4x12s, so she gave me 2x4x16s at the listed 2x4x12 price). She also gave me some additional discounts due to order size. Can't hurt to ask.
HD and Lowe's seem to have fairly different selections on lumber. Some sizes/materials are available at one and not the other. Depends on what you need.
Tools... there can be a BIG difference between the two. You really need to shop both stores and pay attention to what is on sale. If not, you are throwing money away.
Recently, I needed to buy a new air compressor. HD was $50 more ($179 vs $129) for the same exact model. Lowe's was marked "regular price" and HD was marked "new lower price." On the other side, HD was having a sale on a Dewalt drill while Lowe's had the same one for $30-40 more.
On just those two items the difference is close to $100. It adds up quick.
"Big box" stores all depend on people being lazy. They lowball you on some items and bank on the fact that you are too lazy (or busy) to go to the other store. What you save on underpriced product "A" they make up when you buy overpriced product "B." The smart shopper may spend a little more travel time, but the savings can be substantial.