It doesn't sound very scientific, I know, but to me, it's all about the extra oomph the eggs lend to my lather. It just feels richer to me. I like using things that make my lather feel richer, and the more on the natural spectrum those things are, the better.
I don't know if this helps, but from my notes that I jotted down from a different soaping forum, egg whites are mostly water and protein and will behave similar in soap as other proteins like milk and tofu do. The yolks, on the other hand, contribute fats, cholesterol (great for skin) and lecithin (an emulsifier).
IrishLass