Only two large-scale battles between these two radically different types of cavalry occurred: Legnica and Mohi. Both of these battles resulted in extremely convincing Mongol victories, with the Mongols subsequently advancing at will across Poland, Bohemia, and the entire Danube Valley. Only the death of Ogedei Khan in December 1241, and the consequent requirement that Subedei's forces retire to support his claims at the kurultai, spared Europe from further ravishing.
The scale and completeness of both victories attests that the Mongol style of warfare was likely superior to anything in Europe until the late 18th century, its command and control systems orchestrating constant feints, withdrawals and ambushes that so exhausted every enemy that they became easy pickings for the final assault.
A critical key to understanding cavalry warfare is the recognition that each horse only has a small number of charges in him until he requires a lengthy respite for rest and feeding. The tremendous weight of a fully armoured European knight reduced both the length, and the number, of available charges significantly.
Both the Mongol cavalry itself, and their style of warfare, was intended to minimize this exhaustion for their own troops, and maximize it for their opponents. Once the European knights were hors du combat from exhaustion, the Mongols closed in for the kill against the immobile infantry, while harassing the knights just sufficiently to prevent them from recuperating.
The Mongol achieved this through the use of a small horse of great endurance, lightweight armour designed mostly for protection against arrows, and many scores of arrows for each of their mounted archers. An extensive command and control system of flags directed troop movements, and a sophisticated logistical network kept the archers continually supplied with fresh arrows without weighting them down.
No-one the Mongols fought, except themselves, could cope with this combination until their forces fragmented to a size that could be surrounded and overwhelmed by sheer numbers.
Note that the battles of Legnica and Mohi were fought with home field for the Europeans, in early spring (April 9 and 11, 1241) when all the disadvantage of fighting in the hot summer sun in full armour was negated. The campaign began with the forcing of the Verecke Pass in the Carpathians into the Hungarian Plain, about as unfriendly terrain for mounted steppe archers as one can find, and the utter destruction of the defenders under Denis Tomaj.