It will definitely make a difference. Yolks contain various things like fats and emulsifiers that affect the flavor and texture of baked goods.
If you want more details, then tell us what you're baking.
There are lots of people who have a rather simplistic approach to nutrition and think that removing fat and calories makes you healthy. Then they go through recipes for things they want to eat, replace the sources of fat with something which doesn't have fat and doesn't make the result outright inedible, and declare their recipe a success. I think this is what happened here.
In a cake, eggs provide leavening, moisture, smoothness, own flavor, and enhancement of other flavors. Oil provides smoothness and enhancement of other flavors (and possibly its own flavor, if not netural). And while it is not water based, it keeps the moisture in the cake from evaporating, so it makes the cake less dry.
If you are a "simplistic nutritionist" without all this information, you can approximate some of the effects with soda. It will provide moisture, and it will also provide some leavening because it is fizzy. It will provide some flavor of its own too, but frankly, I find the rather chemical flavor of soda to be unpleasant. And it won't have any fat. In the eyes of the simplistic nutritionist, it has successfully replaced the oil and eggs while reducing fat and calories.
From the point of view of a baker, the cake will be a disaster, and won't even deserve the label cake. It will dry out quickly because it has no fat. It will have a bland flavor. Its texture will be terrible. They say "more chewy?" It will miss both the protein structure and the emulsifying agents provided by the eggs. It will be essentially an overwhelmingly sweet quickbread with no redeeming qualities. From a culinary point of view, it will be terrible.
Bottom line: under some assumptions, it is a good substitution. For me, these assumptions are so far from reality as to be useless. It is a terrible substitution.
Best Answer
If you truly desire a white cake, you don't want to make this substitution.
The reason the white cake mix directs you to add egg whites only is to to avoid the yellow color from the yolks, which will tint your cake yellowish.
Cake mixes are highly engineered wonders, and are very tolerant of almost everything a home cook can do to go wrong within reason. Still, substituting in the yolks increases the amount of fat—and more importantly, the emulsifier lecithin, and so will change the texture of the outcome. This will probably not be bad, but still will not be the optimal result from your mix.