Organizations like the Nonhuman Rights Project have filed lawsuits seeking "personhood" for chimpanzees and elephants, arguing they possess the cognitive complexity to deserve basic legal protections.
Technology is the wild card. (cultivated meat) and precision fermentation (dairy without cows) bypass the welfare/rights debate entirely. If you can eat a steak that never had a nervous system, the question "Welfare or rights?" becomes irrelevant. Singapore and the US (Upside Foods, Good Meat) have already approved cultivated chicken.
The core difference lies in the ultimate goal of the protection.