What you find in a big box is definitely lower quality, the question is, will you, the end user, notice the difference or care? If you care, then you don't shop at a big box for the things that matter most to you. For generic needs, a big box will be just fine.
Big box yard equipment, all of it is made by MTD, so those once trusted brands, such as Toro, Cub Cadet, Murray, etc., are literally all the same with slightly different body work and paint. Look up MTD for yourself, you will see they own just about all of the consumer grade brands, and they do not support anything that was built previous to their acquisition of that particular brand.
Lumber, H.D. lumber is a much lesser quality than what you will find in an actual lumberyard. Boards have more knots and more sap running out of them. I find that the lumber is often deformed, a lot of wane or other damage is present as well. A #2 pine board is actually a #3 or #4 (lower grades), again, more knots, sap, and defects than what you will find in a dedicated lumberyard or quality material supplier. Wood fencing is definitely a lower grade than what you will find elsewhere, and now they're stocking a lot of fence material that is basically pulp wood that has been stained to look like a higher grade/quality of material.
The business practices and ethics of H.D. are questionable at best as well, if that sort of thing shapes where you spend your dollars.