Since we consume a fair amount of dairy, it certainly isn't Paleo, though I love to read Paleo sites and often use Paleo recipes. The differences between Paleo and Primal diets are more subtle. Atkins, even though it's also low carb, stands out on its own.
Here is a story from Mark Sisson's site that succinctly describes the difference between Atkins and Primal diets:
With Atkins, you don’t fundamentally change your eating habits; you substitute low-carb products for what you normally eat. Low-carb bread, low-carb ice cream, low-carb snack bars etc, all loaded with fake factory ingredients and sugar alcohols. Over time, you drift back to the real crap and end up back at the beginning.
With Primal, you learn to eat real food and you learn to like real food. You learn why the crap food is crap and you lose your taste for it. You make a real fundamental change and you understand why.