Why is whole grain bread more nutritious than white bread? There are a few reasons.
For starters, because the whole grains are less processed, they take your body longer to digest. This means the carbohydrates (sugars) from the bread are absorbed slower, which minimizes the effect it has on your blood sugar levels. Contrast that to white bread which as Glycemic Index (GI) value that's actually higher than white table sugar! (GI of 71 for one slice of white bread versus 63 for two teaspoons of white sugar)
Since the whole grains are less processed, they also contain more fiber. That leaves you feeling full for longer, which often times, means you eat less frequently. So even though the bread's calorie count is similar, it's the reason why a whole grain diet is recommended for weight loss and diabetes.
Lastly is something that is rarely discussed in reference to whole grains... their antioxidant content. The ORAC value, which is a measure of a food's total antioxidants, tends to be higher on unprocessed grains such as this bread when compared to their processed counterparts (i.e. white bread).
USDA Database for the Oxygen Radical Absorbance Capacity (ORAC) of Selected Foods, Release 2 - Prepared by Nutrient Data Laboratory, Beltsville Human Nutrition Research Center (BHNRC), Agricultural Research Service (ARS), U.S. Department of Agriculture (USDA) - May 2010