The primary feed types are pellets, crumbles and mash. There are also feeds that are formulated from more whole seeds, giving the appearance of being more wholesome. (not necessarily the case)
In answer to your question.
They are essentially the same. The formulas are the same for the same given feed type. The heat applied, usually as steam is to prepare the legume content to be utilized as food, removing the antinutritional constituents common in most uncooked legumes.
Mash doesn't have a binding agent. Pellets and crumbles do. To make product other than mash, the mash is mixed with a binding agent and run through a pelletizer.
To make crumbles, the pellets are run through a crumbler.
One needs to look at the nutritional requirements of the animal fed.
Not just the species but also the age life stage at which animals (and people) have different requirements for optimal growth and development.
That nutrition doesn't come from vegetative sources alone. Chickens are omnivores. Left to their own devices in nature, they will eat a variety of vertebrates and invertebrates. Virtually anything that moves. From those sources, they get the range of essential amino acids necessary. Young birds, whether chickens, other galliformes and virtually all species require higher protein. That doesn't come from vegetative sources.
Most feeds have at their core a grain and a legume. These complement each other in their amino acid makeup but are still not complete. It is decided what the nutritional level needs to be for the animal being fed. For instance, chicks need a higher protein, usually starter or grower feed, sometimes an all flock type feed (also higher in protein than layer or finisher feed. To make up the amino acid deficiencies of the grain and legume, synthetic Lysine and Methionine are added.
Higher levels of vitamins and minerals are needed for optimal growth and body processes as well as fats. These along with the essential amino acids are added as powders mixed with the ground primary ingredients.
Over time, some of these powders separate from the feed. When feeding pellets, you end up seeing fines at the bottom of the feeder after the pellets or crumbles have been eaten. These fines contain all the goodies that are added as powders.
Another thing to consider is that mash usually comes from independent feed mills because they don't have the specialized pelletizers and crumblers of the big mills. All they have are grinders, hammermills and mixers. Mash is also often fed to cage layers.
Mash and even crumbles can sometimes cause issues because evolution has made chickens and most birds able to eat a variety of foodstuffs, not just a steady diet of powders. The crop and gizzard are parts of the anatomy that accommodate that variety. That's why a chicken can eat forbs, a whole mouse or large grasshopper as well as all size and hardness of seeds. The crop will expand to accept whatever the chicken can swallow. Without hands, teeth or knives and forks, the gizzard breaks down the larger, harder items aiding digestion.
Hope this helps. If any confusion persists, just ask.