I will not particularly get into most of the points of your question, as things like nutrition is off topic an highly polarizing and things like taste or in my view almost entirely opinion based and personal preference or culturally based. OK, in violation of that I will make statements about nutrition which is based on years of research while raising eggs, but primarily to give info, not to say do this or this is healthy, except I will start with one opinion: An egg is naturally nutritionally almost a perfect food, containing everything needed for the healthy development from two-cell to a healthy, self-reliant bird through the first week out of the shell. We do not however eat natural eggs, we eat eggs from selectively breed, unnatural birds that are fed a diet very different from what they evolved to eat and have altered the make-up of those eggs in doing so. Our generic chicken eggs stopped being that perfect food long ago, but truthfully, good sourced ones really are not as far away from that ideal as some would claim. Sorry about that, but that was my personal two cents on the off-topic part.
The difference between the yolk and white really boils down to the purpose of the two. The yolk is the main natural nutritional element for a growing embryo. It has most of the things needed to grow a bird and even provide it all the nutrients needed for the first few days of its life out of the shell. It has the fats, yes, including the dreaded cholesterol. In a hen raise with adequate sunlight, a well-balanced diet, and especially allowed to forage, it is actual a well-balanced “good-bad” cholesterol ratio and nothing of concern to those without other cholesterol issues, but that subject also becomes polarizing so I will stop there. It also has saturated fats, but really, we are actually talking about roughly one and a half grams per egg, so probably not a lot as compared to what most of us get from other sources. The yolk contains about 99% of all fat in the egg and the vast majority of some major nutrients like Iron, Zinc, and major vitamins and minerals.
The white of an egg is primarily a protective organ for the embryo and water reservoir. It also contains more of the protein than the yolk, and some vitamins and minerals that are needed more in the later development such as for bones, like Niacin, Phosphorous and Magnesium. Downside for some, it also contains far more sodium than the yolk.
Most of those generalities are from USDA nutritional charts that can be found quoted all over, here is one site that reproduces the chart: https://www.ahealthiermichigan.org/2011/10/11/the-nurtional-value-of-egg-whites-versus-egg-yolks-what-do-you-use/ Always remember that those numbers can readily be altered by changing the diet of the birds though, those are averaged numbers. Is it healthier to only eat egg whites, or only yolks? Well, that is debatable according to your own goals. If you do not want the fats, they exclude the yolk, but you will also be losing a lot of very valuable nutrients. If you have those covered, then your call. Exclude the whites and you are saving yourself sodium, some extra protein and losing some other vitamins and minerals.
The role of eggs in cooking is all over the board, for proving body and structure, an emulsifier, color, flavor, fat, moisture, and on and on and the yolk and white serve different roles in different applications. For instance a classic sponge cake, the yolk and its fats are a binder and help provide the body of the cake, hold it together, while the foam made from the white becomes almost a leavening agent, creating the airiness as the water from them bake out. The roles though can be all over the board and really is too broad to touch on without talking about individual applications in my opinion.
Sorry, too long non answer, meaning the entire question is likely too broad to be on topic.