Biggest difference between "Purebred" dogs and "rescue" dogs is the price, origin, and owners pride. Surprising to most is that AKC registration ONLY indicates the dogs owner and the ancestry, and has NOTHING to do with the dog's quality, soundness, or suitability. Careful and selective breeding are supposed to produce healthier, quality dogs that more closely conform to the recognized, and/or defined qualities of the breed. In the past, this has mostly been true, but price-only produces no guarantees.
For a companion and pet, there is a never-ending supply of healthy, sound, and loving pets available either free, or at a low cost to offset the expense of caring for them. Many of these are excellent dogs, and offer love and affections to those who truly return that affection.