Natural Balance is definitely better then Nutro. However, neither food is that great compared to the higher quality foods like Innova, Wellness, Fromm, etc. Here are the ingredient comparisons -
Nutro Max:
-Chicken Meal (Good!)
-Wheat Flour (should not be in foods, no nutrition, filler)
-Ground Whole Wheat (low grade filler, causes allergies
-Rice Bran (good for digestion)
-Poultry Fat (bad as the type of poultry is not identified - means the company is hiding something)
-Corn Gluten Meal (the dried residue left over from processing corn, low grade filler)
-Ground Rice (good but should be left whole)
-Lamb Meal (good)
-Dried Egg Product (egg shells or eggs not fit for humans)
Natural Balance Original:
-Chicken (good, but meal is better as it has had the water removed meaning more meat in the food)
-Brown Rice (great)
-Duck (good)
-Lamb Meal (good)
-Oatmeal (good)
-Pearled Barley (good)
-Potatoes (good)
-Chicken Fat (okay)
-Tomato Pomace (tomato skins, okay but better if the whole fresh tomato was used)
Innova:
-Turkey (good)
-Chicken (good)
-Chicken Meal (great)
-Ground Barley (good)
-Ground Brown Rice (great)
-Potatoes (good)
-Natural Flavors (good)
-Ground White Rice (good)
-Chicken Fat (okay)
-Herring (good)
-Apples (great)
-Carrots (great)
-Cottage Cheese (good)
-Sunflower Oil (good)
-Alfalfa Sprouts (great)
-Egg (excellent)
Innova is the healthiest pet food in the world so you can see when comparing Natural Balance and Nutro to it that Natural Balance definitely comes closer to resembling the ideal ingredients for a pet food.
If you are thinking about switching to a higher quality food then it would be my first recommendation that you go with Innova because it is the better of the three however, Innova is not sold at Petco or most mass suppliers. The company likes to put its food into smaller stores where the staff tend to be more knowledgeable about food. If you check out their website www.naturapet.com you can find out if there are any other stores near you that sell it.
If you can't find it then by all means go with the Natural Balance as your dog will benefit from the added nutrition.
*My only other piece of advice that I like to pass on the people (and I mention it because of another post to your question) - VETS do NOT receive adequate nutrition training and are therefore not able to provide their clients with true beneficial nutritional advice for their pets. Most vets prescribe or recommend foods like Eukanuba, Science Diet, Purina, Royal Canin, Walthams, etc and these foods are all lower quality then the Nutro. So for the best nutritional advice, find a small pet specialty store with friendly knowledgeable staff and also take the time to check out ingredients yourself.
All the best and good luck choosing a new food.