Philosphy: Atkins is simply a "diet" intended to help people lose weight. Primal/Paleo is about achieving superior health and weight loss is more of a side effect. There is also more to the primal lifestyle then diet, but I'll just start with the diet differences.
Dif about carbs: Technically you could have grains on atkins and still stay within your carb range. You can also do a more moderate carb version of paleo/Primal based on your activity level. It would just that these carbs would need to come from fruits/vegetables and not grains (with an exception of white rice) Atkins does not have variations.
Dif about veggies: There is also a much heavier emphasis on eating your vegetables on primal/paleo. It's important that we are getting all the nutrients we need.
Dif about dairy: Paleo said no dairy, primal limited dairy if it doesnt bother you and raw full fat, atkins as long as you are in your carb range.
Dif about fat: Atkins also does not care what kind of fats you eat.. Primal/Paleo cares a lot about this, great emphisis is on acheiving an ideal O3/O6 ratio.
Processed food: always a no-no on paleo/primal...atkins again the only thing that matters here is carbs
Primal also discusses other lifestyle choice such as quality/quantity of sleep, type of exercise (which is critical for health!), how to manage stress, getting in touch with nature... and more.
I would recommend buying and reading Mark Sissan's book.
Primal since March 2011
Female/29 years old/5' 1"/130ish lbs