Weston A. Price Foundation (WAPF) "style" nutrition, like from the Nourishing Traditions cookbook, is broadest, as it includes grains and dairy. It focuses on traditional foods, including non-processed foods. You can read more about it here: http://www.westonaprice.org.
Paleo or primal eating focuses on nutrition as our "caveman" ancestors would have eaten, particularly meats, fats and some berries and seeds. It is very healing, as well, but doesn't focus on healing "the gut" in quite the same way that GAPS does.
is similar to Paleo, but GAPS is geared toward healing the body through
focusing on certain foods and avoiding others. Think "nutrient dense."
There are various stages to the diet. A lot of people to the "full" GAPS diet, then move to the stricter healing stage of the "introduction" diet. It also
incorporates an elimination diet of sorts. The "full" GAPS diet is
less strict than Paleo, which really surprised me.
Also, this "cheat sheet" for GAPS is useful.
GAPS and Paleo are to be consistent with Nourishing Traditions, but not all Nourishing Traditions food is consistent with GAPS and Paleo.