Atkins restricts all forms of carbs during the induction phase including veggies and fruits. Atkins puts no priority on quality cuts of meats or good oils to use for cooking.
Paleo is how many nutrition gurus have interpreted the way we might have eaten prior to the agricultural revolution. Whole foods, meats, veggies, fruits. Paleo is normally pretty strict and some people don't use dairy. Some paleo gurus like Cordain and DeVany recommend leaner cuts of meat. While others recommend fatty cuts of meat just as long as they are from good grass fed sources. n3:n6 ratios are somewhat monitored and keeping inflammation down is key. Chronic exercise is normally avoided because of the inflammation factor.
Primal is Mark Sisson's interpretation of paleo. Primal incorporates more dairy if its tolerable. The main points of paleo are adhered to, whole fresh foods, meats, veggies, and avoiding grains.
"If man made it, don't eat it" - Jack Lallane
People say I am on a "crazy" diet. What is so crazy about eating veggies, fruits, seafood and organ meats? Just because I don't eat whole wheat and processed food doesn't make my diet "crazy". Maybe everyone else with a SAD are the "crazy" ones for putting that junk in their system.