This is taken from marksdailyapple forum.

Atkins has four phases:
1. Induction: you drastically reduce carbs. Suggested range is net carbs of under 20grams per day. A net carb is calculated by taking total carbs grams and subracting fiber grams. It’s recommended that you only stay on this phase for two weeks. Phase one really limits produce choices to leafy greens and other similar low carb choices. No fruit is allowed. No caffeine is allowed (the idea is that caffeine inhibits your body’s ability to enter ketosis).
2. Ongoing Weight Loss: You slowly increase carbs, by about 5 grams per week. You add in more of what are considered foundation vegetables, can resume caffeine intake, and can introduce non-starchy fruits.
3. Premaintenance: Your weight loss will slow down. Once you’ve reached your set point, or the point where your body stops shedding weight, that is considered your carb threshhold. At this point you should be about 10 pounds from goal weight.
4. Maintenance. You maintain eating according to your carb threshhold to maintain the weight loss.

Atkins doesn’t place much emphasis on exercise, sunlight, or levels of toxins in food. You won’t get much information about benefits of grass fed protein, free range eggs, raw dairy, or direct sunlight from Atkins. Primal is a lot more wholistic, and not necessarily focused on being low carb, although you can make it low carb if you want.