How is Paleo Diet different from Atkins' Diet?

The biggest difference, in practice, is around refined seed and vegetable oils. In Atkins they are okay, whereas in Paleo they are a big no-no.

Atkins revolves around the reduction of carbohydrate intake. Paleo revolves around eating natural pre-agricultural foods. Atkins is low-carb. Paleo is usually, but not necessarily, low-carb. Refined seed and vegetables oils are okay in Atkin's since they are low-carb. They are not okay in Paleo since they are neolithic frankenfoods. Paleo prefers grass-fed beef and pasture-raised poultry over their conventionally raised counterparts. Atkin's doesn't make such distinctions.

So, something like mayonnaise would be okay on Atkins but not okay on Paleo. You can find "Atkins-Advantage" granola bars in stores; such things are completely preposterous from the Paleo perspective.

A disclaimer: I know Paleo pretty well, but I have not studied Atkins in detail, so my answer may be incomplete.