Atkins stresses low carb. Period. It doesn't seem to have much interest in real food.
Paleo stresses real food, removing foods that don't nourish you, increasing foods that do, and whether that happens to be low-carb or not is your choice/needs.
Just an aside...Atkins can totally be done paleo-style. There is nothing about Atkins that requires you to eat grains, legumes, dairy, and certainly not processed, boxed, Atkins-brand or any other brand of food. Many may do it that way, but it is certainly not required, and in the earlier Atkins books (the ones written when Dr Atkins was still alive), I believe whole foods are encouraged/preferred
The main gist of Atkins is that you are eating below a certain level of carbs. You are supposed to start off with a two week induction phase where carbs are kept below 20g net carbs (carbs minus fiber grams). You can eat protein in the form of meat, eggs, dairy (dairy is not required, and in fact limited during induction). Grass-fed, pastured meats, etc. may certainly be chosen. You are not supposed to eat fruit, nuts, or starchy veggies during induction, but you can have several servings of non-starchy veggies a day (in fact, I believe true Atkins makes this a "requirement"). After the two weeks are up, the idea is that you have probably lost your cravings for sugary, high-carb foods. Those with a lot of weight to lose may choose to stay on induction for a while, while others can choose to go up the "carb ladder" one week at a time, adding in both variety of carbs and amount. I think it is an increase of 5 carbs per day that you add for a week at a time (so the first week after induction would limit you to 25 daily net carbs, the next week 30, and so on). You do this until you cease losing weight, and then you drop back to the previous level and stay there until you lose what you want. There is also a prescribed order of adding back variety, I forget exactly...I think berries may be first, nuts next, etc.
Ok, probably more than you wanted to know, but I came to paleo through Atkins. Atkins got me into researching the baloney of the "heart healthy lowfat diet," and started me on my weightloss journey. Over the years my research has led me to paleo as a healthier way to eat. I know that I personally do better on a lower carb eating plan, at least for now. You can definitely do Atkins and Paleo at the same time. Just remember to stay away from the processed junk, and yes, to echo above. Do not go low fat :)
hopefully this helps!
In slightly more detail:
Atkins: Don't eat more than X grams of carbs, but otherwise do whatever you want.
Paleo: No grains, legumes, dairy (except butter and cream, depending on your specifics), sugar, vegetable oils (olive and coconut are fine). Prefer grass-fed and pastured meat over factory farmed meats and eggs. There are more details, but this would probably take care of you for a few years.
People who want to lose weight often combine the two, with low-carb paleo. Athletes sometimes fuel their workouts with lots of carbs, e.g. sweet potatoes.
Oh, one last point. They sound hard and boring. Once you've done either for a month, you'll lose your sugar and bread cravings, and suddenly the world of food is your oyster. Sounds implausible, but it's true. There's nothing like eating as much as you want without worrying about it making you fat.
If you do either of these, don't think you'll "have the best of both worlds" by going low fat too. That won't work. You'll have to eat something, and it can't all be protein.
atkins wants you to eat atkins brand products that are low carb. i couldn't read the first 2 ingredients, let alone the other 30 ingredients which i assume to be flavor producing chemicals.
paleo doesn't need to be low carb, doesn't need nutrition labels as there shouldn't be more than one ingredient (assuming your cooking at home).
eat paleo for a week then try an atkins brand product- you can taste the chemicals!
Weight loss on Paleo--HOW? 17 Answers
Potatoes and feeling good? 2 Answers