From what I know, the boots were of the type normally called half-boots and were nicknamed "Bluchers", though I don't know the story behind that. Shoes are normally considered to be footwear that do not enclose the ankle. Boots of the time were best suited for riding and were needed to protect the leg from friction agains the the animal and from being torn when riding through brush. Boots for riding could range from fairly light weight to very heavy (consider hussar versus cuirassier boots). Personnel expected to be on and off their horses, such as staff officers and the mounted officers in infantry battalions favoured light cavalry boots for obvious reasons. Apparently Wellington requested his cobbler to modify the standard Hessian type boot. He wanted a lower heel and to have them cut lower down the calf. The typical design was more form-fitting and had a deeper heel to aid in holding the stirrup. Since Wellington spent much time at a desk, it seems he wanted footwear suitable for all purposes. The boot he had designed for his purposes had nothing to do with today's rubber rain boots. The original Wellington boots were a hybrid riding boot and were not intended for infantry use.
As far as I can tell, those 1823 boots were low calf height and intended to protect the ankle. In appearance they might be similar to hiking boots today. I think they were lace up, whereas riding boots were not. Lace up shoes had become more common during the wars (as opposed to buckle or slip on). Obviously slip on shoes were the cheapest, but of only limited use. Lacing shoes required some form of eyelets, which in turn required more material and more work. The eyelets were probably of the style that had been around since Roman caligulae and still found in dance shoes, as opposed to the individual metal eyelets most of us are used to in our shoes and boots.
Leggings and breeches had survived for so long for reasons of economics and how the legwear was intended to be used, as far as I can tell. In the civilian world, there is no connection between footwear and legwear except to coordinate for fashion. In the infantry there are direct functional connections. The most basic is that it is important to cover the tops of the footwear to prevent material falling inside the tops of the boots. Such material can cause blisters and other damage with remarkable speed. It is not acceptable for individual soldiers to sit and clean out their boots whenever they feel like, so real injury could develop. Shoes were relatively cheap, but very low cut. Breeches were higher cut. Therefore they needed some kind of overlap. This website: http://www.warof1812.ca/trousers.htm shows a great picture of gaiters overlapping the shoes. By going to boots, cost per unit would go up, but they were probably repairable for longer. I also reckon that industrialization made it possible to produce boots more cheaply than before. I can tell you as an long term infantryman that footwear and legwear was of extreme importance to us.
You'd need to speak to an expert in the period between Waterloo and Crimea to validate the above, but hopefully it gives you a few lines of inquiry to follow up. Regards,