One difference is that the "new" Atkins indicates you should eat 20-25 NET carbs per day (which is total carbs minus Fiber and Sugar Alcohols). The "old" Atkins, I believe, indicates that you should eat 20-25 TOTAL carbs per day (so you don't subtract fiber or Sugar Alcohols).

The Rules of Original Atkins

The Diet Revolution Rules (Level One)
1. Don’t count calories
2. Eat as much of the allowed foods as you need to avoid hunger.
3. Don’t eat when you are not hungry.
4. Don’t feel you must finish everything on your plate just because it is there.
5. Drink as much water or calorie free beverages as thirst requires. Don’t restrict fluids… but it is not necessary to force them either.
6. Frequent small meals are preferable.
7. If weakness results from rapid weight loss, you may need salt.
8. Everyday take a high-strength multivitamin pill.
9. Read the labels on “low-calorie” drinks, syrups, desserts,. Only those with no carbohydrate content are allowed.

The First Life-Changing Week, eat nothing that is not on the list.
Meat: Steaks, Corned beef, Lamb chops, tongue, Hamburgers, Bacon, Any kind of meat in any quantity—except meat with fillers such as sausage, hot dogs, meatballs, most packaged “cold cuts”
Fowl: Duckling, Turkey, Chicken, Anything with wings, No stuffing
Desserts: Gelatin with artificial sweeteners (e.g., D-Zerta [is that still around?])
Condiments: Salt, pepper, mustard, horseradish, vinegar, vanilla, and other extracts; artificial sweeteners; any dry powdered spice that contains no sugar
Drinks: Water, Mineral water, Vichy, Club soda; Beef or Chicken broth, bouillon; Sugar free diet soda; coffee*, tea, decaffeinated coffee

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*Special note on caffeine and Diet Cola. Because most heavy people have some hypoglycemia, coffee, tea and diet colas, which contain caffeine, should be limited to six servings a day (cups). If you know you have low blood sugar, better limit it to three.
Fish: All fish, including canned salmon, tuna; any kind of seafood, including oil-packed and smoked, except oysters, clams, muscles, scallops, and pickled fish
Eggs: Boiled, fried, scrambled, poached, omelet – any style and without limitations
Salads: Two small green salads a day (each less than one cupful, loosely packed) made only of leafy greens, celery, or cucumbers and radishes. Dressings with vinegar, oil, salt, dry spices, herbs, grated cheese, or anchovies. Or else a sour pickle in place of a salad. Plus… green olives.
Butter & Mayonnaise Fats: Butter, margarine (head New Diet Revolution & eat no margarine with trans fatty acids), oils, shortening (not such a great idea), lard & mayonnaise (fats have no carbohydrates).
Juice: Juice of one lemon or lime
Cheese: Four ounces a day of any hard aged cheese. No cream cheese or cheese spreads.
Heavy Cream: Four teaspoons a day. (Cream has less carbohydrate than milk – so don’t use milk)
Diet Revolution Salad Material: Celery, Chicory, Chinese Cabbage, Chives, Cucumber, Endive, Escarole, Fennel, lettuce, Olives (Green or Black), Onions, Pickles (Sour or Dill), Parsley, Peppers, Radishes, Scallions, watercress.
What Happens After The First Week’s Diet. Most diets are fixed formulas and are therefore short-termed and self-limiting. This diet is fixed only for a week. After that – because it must be a lifetime diet, it is, as variable as are individual tastes.
I’ve labeled the additions of carbohydrate that follow’ second level’, ‘third level’, ‘fourth level’, ‘fifth level’, but in fact the additions are interchangeable and flexible. You can make any of these additions any week that you choose. I am so committed to making this a livable lifetime diet that I am letting you select your own variations, within the rules set up by your biological rulebook.
Put back what you’ve missed the most: The idea is simply to gradually return to your diet first what you missed most. Custom-tailor the diet to suit your carbohydrate additions to suit your lifestyle. All that matters is that you add back to your diet a little carbohydrate at a time, and that you stop adding carbohydrate when you reach your CCL.
How to know when to put back a little carbohydrate: Ask yourself: Are the keto-stix still turning purple? Am I still un-hungry? Have I stopped eating at night? Do I have more energy? Am I still losing weight or inches nicely? Remember, your tape measure is a lot better friend than your scale, not only more accurate, but better able to report on the actual fat (not just temporary water) losses this diet achieves.
The Second Level: Cheesecake for dessert? At each level, remember you add approximately five to eight grams of carbohydrate daily for a week and analyze the results. Most people agree the best way to handle the second level is to add cottage cheese…
The Diet Revolution Vegetables (not for level one): All of the above plus; Asparagus, Avocado, Bamboo Shoots, Bean Sprouts, Beet Greens, Broccoli, Brussels Sprouts, Cabbage, Cauliflower, Chard, Chinese Cabbage, eggplant, Kale, Kohlrabi, Mushrooms, Okra, Onions, Peppers, Pumpkins, Rhubarb, Sauerkraut, Snow Pea Pods, Spinach, String Beans, Summer Squash, Tomatoes, Turnips, Water Chestnuts, Wax Beans, Zucchini Squash