I am not a nutritionist, nor am I an expert, however, I have asked many people of many walks of life what the difference, and the only common(which is usually in most cases the correct) answer, is read the label. the ingredient list on a label is list in the order of the amount of each ingredient from highest to lowest. Whole grain breads are typically healthier, because they off more grains than whole wheat bread. if the label says bleached, processed, or refined, than it isn't that much healthier because the process strips almost all of the nutrients.
Always read the labels on anything you eat. I learned that in my nutrition class in college.