What advantage did Mamluk horses have against Mongol horses?

Answer 1

General Answer
NONE. Mongol horses were better trained and hardier horses than those used by the Mamluks. The reason that the Mamluks were able to prevail at the Battle of Ain Jalut was due to a better knowledge of the terrain, better tactics, secret alliances with the Christians that the Mongols were unaware of, superior numbers, and that the majority of the Mongols and their leader Hülegü Khan had recently departed from the front line for internal reasons in the Mongol Empire. While it is the case that the Mamluks used their horses to perform quicker maneuvers in the course of the battle, Mongol horses could have done those same moves, the Mongol riders simply chose not to employ that tactic.

Specifics on Why Mamluks Won
While better horses were not part of the reason for Mamluk victory, it is worth bearing down on some of the actual reasons for the Mamluk victory to understand why horse ability was irrelevant.

Immediately prior to the Battle of Ain Jalut two major things occurred that led to a fundamental change in the skill level and troop numbers on each side, negative for the Mongols and positive for the Mamluks: (1) Great Khan Möngke died and Hülegü Khan, the leader of the Mongol Forces in the Middle East and Möngke's brother, was required to return to Mongolia for a kurultai (vote for the next Great Khan) and brought the majority of his forces with him, leaving a smaller force under his lieutenant Kitbuqa. (2) Kipchak Türks fleeing west Turkestan (modern day west Kazakhstan) because they were a group that did not want to surrender to the Mongols found refuge with the Egyptian Mamluks and bolstered their numbers significantly.

When the battle started, the Mongols were caught off-guard, primarily because they did not expect for the Christian Crusaders to permit the Mamluks to cross over their lands to launch an attack on the Mongols, given the Crusaders' long abiding hatred of the "Saracen Mamluks". Once the battle had commenced, Baibar, the Mamluk general, used his extensive knowledge of the terrain in crafting his tactics and coordinating the battle. The battle had two major components; it started with a hit-and-run, where Baibar targeted Mongol cavalry and finished with a feint. A feint is where an army appears to retreat in defeat and the enemy pursues, usually in an attempt to rout the retreating army. However, the retreat is fake and leads the pursuing army into an ambush which they cannot handle as effectively since they are out of formation. This is exactly what happened, where after the Mongols pursued the falsely retreating Mamluks to the highlands, Mamluk forces emerged from hiding and began to fire arrows and attack with their cavalry. The Mongols were surrounded and defeated.

Answer 2

They were fitted with horse shoes