Giant Eagle and Giant are two totally different stores. I've never been to a Giant, mostly because there's none in this area. But I've heard of Giant before.
FYI for Erie-ites, the Yorktown Giant Eagle is a franchise, while the rest are part of the chain. The only reason I mention this is that they carry some different products and not necessarily everything the other stores do.(especially with specialized products: low-carb, lactose free...)