I have been reading the website and I just got the book. The Primal Blueprint and Atkins are almost identical. I would say about 90-95%, so there is not much I would change in order to "go Primal".
They both teach high fat, moderate protein and controlled carbs. They both teach against eating processed carbs, like sugar, flour, etc. They both teach to eat in the same calorie range.
You have to look pretty hard to see even minor differences. Here are some minor differences, as I see them:
Atkins starts with around 40-50 total carbs (20 net carbs) per day to get into ketosis. PB starts with 50-100 total carbs per day and is not necessarily ketogenic. But PB does say that you can stay under 50 total carbs per day so that you can enter ketosis for faster weight loss if you choose to. Mark Sisson does not say that ketosis is wrong. He believes that ketosis was the way humans were designed to burn fuel.
Atkins says to eat at least 3 regular meals per day, if not more, and that if you skip meals, you will or may slow down your metabolism. However, the new book does say that some people do better with less than 3 meals a day. PB says to eat when you are hungry, and that might only be twice a day, without risking a metabolic slowdown.
Atkins emphasizes diet and talks less about exercise, although it is encouraged. PB emphasizes diet and exercise equally, and there is a lot of info on doing low level cardio, and stopping the "Chronic Cardio".
Atkins has options for eating some processed foods, although they recommend eating real, whole, fresh food as much as possible. PB has no options for eating processed foods. To do it right, you must eat only real, whole, fresh foods.
Atkins says to eat organic and grass fed, etc., when you can, but you can still do Atkins "right" if you eat non-organic, grain fed, etc., foods. PB says to eat organic and grass fed, etc., and this is necessary to do PB "right".
So all in all, aside from a few minor things, I don't see anything that different in PB. It sounds like Atkins, if you want to take it the extra mile and eat all organic, grass fed, etc. It's the way I have been wanting to do Atkins from the beginning, but could never quite get the gumption to follow through.
Just for the record, I will always recommend that people follow Atkins here on the forum. If I choose to fine tune Atkins a little within recommedations, that is my choice and I would never push that on anyone else.
For me, personally, I feel that Atkins has been what I have needed to overcome my eating problems. I just know that I am ready to take it the the next level of eating organic, etc. I don't see it as stopping Atkins and starting PB, or something else new. The way I see it, this is still Atkins, and I still really believe that Atkins is a great way to go.