The steppe pony is smaller than the horses that the Europeans were using and didn't subsist on grains like stable horses. They were also known for their endurance, hardiness, along with not needing to be "coddled".

This meant that the horse would be more maneuverable in combat, could subsist merely on grass, and last longer than European horses. However they weren't as fast, which was it's main disadvantage.

European horses is a large group to choose from. This ranged from the small ponies that many countries in Northern Europe preferred to the larger horses that carried knights. Arabian horses too were introduced in Europe by the time of the Mongol rise thanks in part by Muslim conquests in Iberia and the Crusades. However for this purpose I'll look at warhorses like the Destrier compared to Mongolian horses. Mongolian horses, as pointed out in other posts, were a small stocky breed and were also very hardy. They didn't require the same maintenance of European horses as they were raised in the extreme climates of the steppes and could rely on simply grazing on grass and foraging for their own food, reportedly even in the snow and eating leaves from trees. This hardiness and their great endurance allowed Mongol armies to travel great distances without large baggage trains (some Mongol soldiers preferred lactating mares to make use of the milk which they used for foodstuffs included a fermented beverage). However horses like Destrier had two clear advantages: speed and size. While Mongol horses were probably more maneuverable, in a straight sprint, their stocky legs did them no favor. This greater size and speed gave a greater force at the point impact. This reflects the nature of warfare between Europe and Steppe people: European armies were primarily infantry based and the cavalry that developed there was used to crush/harass (see light cavalry like Jinetes) enemy formations and were often seen as a part of an elite arm of the army and in many cases the social order as well. Horses for the people of the Steppes were common and weren't seen as a luxury so warriors often had a collection of horses to choose from, so warfare evolved to take advantage of access to horses and techniques originated to deal with cavalry based armies.

While I'm not as well versed with East Asia, the Middle East was the birth place of the Nisean horse a now extinct breed that the empires of Classical Persia made use of including the cavalry heavy armies of the Sassanian and Parthian empires. The Arabian horse is probably one of the oldest breeds in existence, and were a force to themselves noted for their speed, agility, and great endurance and acclimation to the desert.

Research undertaken by Museum of London, using literary, pictorial and archaeological sources, supports military horses of 142-152 cm, clark, p25, while the destrier were not much bigger at 152 to 157 cm. The medieval Scandinavian horse was between 120 and 126cm [Kulturhistoriskt lexikon för nordisk medeltid].

I'm personally interested in medieval Scandinavian history, so I'll give my account of horses used there.

The Scandinavian horse was wildly used in warfare as land transportation in what was often fought during winter time (Oslo siege 1308/09, Värmtland campaign 1225, etc.). The lakes were frozen which allowed the army to quickly cover ground which would otherwise be covered with swamps and marshes in the summer. There is no descriptive evidence to support that native horses were used for knightly couched lance tactics. Instead, we see a lot of use as scouts, mounted bowmen ("dragoons"), allowing the troops to get to battle well rested or get away from loosing battles.

We see evidence of tilting lances in Scandinavia only in a tournament setting for royalty (DN II 140, Old Norse), German mercenaries knights (Fagerland) and in educational and normative sources (King's mirror, ch 38).

The Mongolian ponies were very small- they're basically unchanged today from what they were 600 years ago. Most are under 14 hands high. They were hard, scruffy, and roughly trained animals.

The European horse breeding program was considerably more sophisticated than anything the Mongolians had. The Europeans were developing things like stud books and breeds, whereas the average Mongolian breed for just speed and endurance. European horses were much taller (at the time, about 15-16hh). This corresponds to a significantly greater stride angle. http://www.somaxsports.com/photo.php?analysis=california_chrome

Basically, stride angle is what allows for speed in the gallop. It's essentially the amount of distance a horse can cover in a stride- and horses with longer legs have a longer stride as well as a bigger stride angle, which allows for a much faster horse.

Europeans were less concerned with endurance in their warhorses, at least in the time of knights. A knight would ride his riding horse to the site of the battle, where he would then mount his war horse. Mongolians would have a string of ponies and use whichever was their favorite or the most fresh when they attacked. Basically, European horses were more purpose built- drafts for pulling, horses for riding, horses for war, horses for racing, etc.

European horse training was also more developed, which comes naturally out of European style mounted warfare- close quarters combat and charges with polearms required very tuned, muscled horses that could be ridden with precision.

A Mongolian horse was much less "trained" than a European horse. Basically, Mongolian ponies are trained to move off the riders legs and weight as the rider shoots his bow and arrow. There were Mongolian lancers, but they were much less prevalent than archers. The ponies were basically allowed to do what they wanted, provided that they basically traveled in the correct direction. Speed was more up to the horse than the rider.

European horses were typically more temperature sensitive than Mongoloan ponies, but things like horse blankets tend to help with than. Regardless, horse combat in snow or ice is never advisable, especially with horse shoes, because the snow balls up in the horses' feet in such a way that they are prone to slipping. Mongolian horses did not have this problem.

Arguably, the finest war horses in the world came from Arabia. The Arab horse is the foundation of almost every European breed- the thoroughbred breed can trace its lineage back to three Arabians imported hundreds of years ago. Europeans imported Arab stallions to build their breeds. The Arab was fast, smart, and had tremendous endurance. They were trainable, strong, and had a good temperament.

The Duke of Wellington's favorite mount was an Arab named Copenhagen. He rode him throughout almost all of his campaigns against Napolean. Likewise, Napolean had a string of Arabians trained as his mounts- they were expertly trained and probably some of the best warhorses to have ever existed.

No other horse really had the impact the Arabian on horse husbandry.