NOTE: This post has been updated on 12/7/16 as a result of a post by Vye indicating that the Thoroughbred would be renamed the Purebred, and giving information on what that means.
As it stands, we have 5 confirmed types of mount. I would like to propose that there not be a flat "X Mount is better than Y Mount" delineation as I've seen some other people talking about.
So, keep in mind that I am considering the EP Store Price when it comes to this, and trying to make sure that people are still getting more bang for their buck out of the more expensive mounts, but without just declaring that, say "A Trison is better than a Horse in every possible way, because it was more expensive."
Ultimately, there needs to be an in-game reason for all of these mounts to exist and be useful apart from 'that one was cheaper,' otherwise the cheaper mounts will start to disappear as the game progresses. After all, if a Trison is better at everything but being cheap, then breeders will focus on them to the exclusion of all else, the larger numbers of them in existence will drive the price down, and now a Trison is fairly inexpensive as well as being the best at everything. That's a bad gameplay model, IMO. And, this is also to give purpose to all of the people who own higher-tier packages and thus own multiple pack/mount animals.
So, my proposal is that different mounts be good at different things. That their attributes reflect what they are and what they can do, as opposed to a Trison simply being "Mount +5." Naturally, Animal Husbandry can mess with this via breeding, but not to the extent of crossing the lines.
All that said, here is my proposal for the distinction between the different types of mount, and what they are most useful for.
Donkey
Donkeys are little, they are chill, and they are sure-footed, and this is a very good thing. While a donkey has a lower load capacity compared to other mounts, isn't very fast, is not terribly durable, and is too small for an adult to ride--they can go anywhere you can and are chill enough to not freak out. If a Miner is going underground, they could take a Donkey with them to help carry the ore back out. If a Mason need to work on a chunk of stone, a Donkey could drag it directly into the workshop. If you don't mind, they can even follow you inside while carrying their load and can navigate stairs and tight spaces just as easily as you can.
EDIT with thanks to Scheneighnay: Donkey are also intelligent enough to not do something that will likely get them hurt or killed. If you are riding a horse and try to steer it off a cliff, it'll go off the cliff. If you're leading your donkey somewhere and it puts on the breaks...it might be a good idea to see what just spooked it.
Horse
A Horse is your general-purpose jack of all trades mount/pack animal. They are middle-of-the-road in basically every stat from carry capacity, to endurance. The one outlier is speed...in a long distance race, a horse will outpace any other type of mount (though a Trison can overtake them in a sprint). If you need an animal that can do basically anything fairly well, look no further.
Purebred
A sub-type of Horse, a 'purebred' is more of a technical differentiation rather than a hard mechanical one. According to this post by Vye, a Purebred is 'better stock' than an ordinary horse. What this means is that they are a horse that comes from a line of horses that has been domesticated for generations, and have been selectively bred for quality and ease of taming.
Given time, you could repeat this process with 'ordinary horses.' So, in essence, a Purebred is a head-start on the breeding process. In-game, this distinction may become blurrier with time as players start selectively breeding normal horses to make their offspring better, or they start mixing normal horse into a purebred line to try to grab some attributes they are lacking. Mechanically, they are identical in utility (jack of all trades animal with a good top speed) to a normal Horse, but are of higher quality. This is one case where I don't feel the need to keep them 'balanced' with ordinary horses because they are simply a head-start on what any breeder could eventually accomplish with horses.
Ursaphant
An Ursaphant is a purely-focused bulk hauling animal. They are slow, but can haul a ridiculous amount of weight, and all that fat makes them quite durable if attacked. Additionally, they are the first of the mounts that will seriously attempt to defend itself even without significant training. The others are more likely to run if they are attacked.
Trison
A Trison is most often seen as an engine of war. They are strong (though not so strong as an Ursaphant), extremely durable, and are ferocious fighters when threatened both by charging their foes and by rearing on their hind legs to pummel their enemies into submission. They are also excellent sprinters, reaching a higher top-speed than a Horse, though they cannot maintain that speed for very long. This makes them terrifying shock-cavalry as they can put on an enormous burst of speed just before they strike enemy lines. That said, a Trison is also excellent in other non-warfare roles--after all, if you have to leave your wagon to go harvest materials, it is quite useful for the animal pulling the wagon to be perfectly capable of killing most things that would mess with the wagon.
Wrap-up
I feel that this sort of layout, where different mount/pack animals have differing strengths, weaknesses, and niches creates a more diverse game. Not only does it keep each type of mount useful throughout the game, it also allows for varied tactics that use different sorts of creature, increasing the variety of what we'll see running around...and giving us a further idea of what to expect when we see someone else.
For example, if/when Mounted Combat comes online...Trisons will likely make-up your Heavy Cavalry, best suited to a straight on, brute-force charge. Some horses, as all-rounders, serve as light cavalry, harassing the enemy's flanks and striking at their supply lines. Other horses, bred for speed and endurance, will carry scouts and messengers swiftly to their goals. Ursaphants tow siege weapons and supply carts. And.....Donkeys are back home, contentedly living their lives, as is their wont.
I frankly think this would be a lot better than everyone who could afford one always hauling around on a Trison, regardless of what they were trying to accomplish.
As always...I welcome comments, additions, opinions, and so on. Let me know what you think!