Neither is the clear winner.
Ditto for all the grocery stores.
I've kept records, and just to give a few examples:
Target is the winner on most paper products, but they are very weak on basic staples for people who cook from scratch, and their mixes tend to be good quality but rather high-priced.
Canned goods tend to be lower at
Walmart -- unless they're on sale at Target.
Target sells some specific things that I can't get at Walmart -- the taco seasoning that I prefer, for example.
Aldi's beats them both for milk and bread, and they have some unique items that we like in their freezer case, but their selection of veggies is sorely lacking.
Harris Teeter's eVIC specials beat everyone, but only for those few items each week. For everything else, they tend to be significantly higher.
Food Lion tends to be the best-rounded, overall least expensive store; however, they sell only food, so if I need to pick up a prescription or something else, sometimes I don't choose them. And in a few odd categories (for example, soup) they're sky-high.
If you really want to pay bottom-dollar for your groceries, read the first edition of The Tightwad Gazette by Amy Daczyn (I know I got that name wrong). She proposes an excellent "price book" that lets you keep track of each store's prices on the basics. It takes a while to set it up, but once you have it, you can track who sells this or that for the best price -- and you know what's a real sale and what's a fake sale. And you'll be able to predict times of year that certain things are likely to go on sale. It's a project that's well worth the time.
Here are two examples:
Having put together a price book, I know that Walmart always has the best price on peanut butter, especially if I buy the huge (is it 8 pounds?) size. So before I go shopping, I check my list of "Walmart's bests" and I see that I'm going to need peanut butter in a few weeks. But I may not go to Walmart again when I actually need the peanut butter, so I go ahead and buy it now. This prevents me from paying more for it next week or the next when I'm actually out.
Boxed mac-and-cheese is an item that's all over the map price-wise. If I just look at Target, I might think that the .99 cent "on sale" boxes are a good deal -- because the shelf tells me that it's usually $1.29; however, my price book tells me that Walmart sells Kraft brand every day in boxes of 5 for $2.50, which is only .50/box.
So, my best advice: Start keeping records. You will be amazed at what you learn, and you'll lower your costs. But it will take you 6 months to a year to really build up the information on your area. This isn't a quick-fix. I'd suggest that you start with one category, say Dairy, and then move on to canned goods. Unless you have massive time on your hands, you won't be able to do it all at once, and you'll just give up.