The simple reason is geography.

For those who don't know what a "steppe" is, the term refers to the flat, uninterrupted grasslands that form a large part of the landscape of Eastern Europe and Central Asia. The lack of resources meant that the people living in these regions were mostly nomadic, but it also meant that they were able to move across the continent without having to cross significant natural barriers. The scarcity of resources would drive the nomadic peoples to migration and war. Key to this expansion was the right horse breeds. The Mongolian horse is significantly shorter and stockier than other breeds, and while slower, they provided a more stable riding platform. The lack of appropriate wood meant that development of the composite bow, a much shorter bow combining animal horn, wood and sinew, resulting a weapon that, incidentally, could be used effectively from horseback. Lifestyle was also a significant factor, as hunting on the steppes would also have been done from horseback. Horses were so important to the steppe people, each warrior basically lived, ate and slept together with his horses, of which he maintained several and alternated between them.

In contrast, Europe was mostly agricultural with large expanses of forests and mountain ranges acting as natural barriers. Without the need to cross wide expanses of grasslands, horses were only typically used for warfare by the elite, and most hunting was done on foot using bows or spears. The wider availability of appropriate woods, such as ash and yew, made for better self-bows that were popular throughout Europe, and crossbows were also widely adopted. Thus, in the geography of Europe, there was little need for a highly mobile army consisting nearly exclusively of horse archers. Most armies were more conventional forces with footmen, supporting missile troops, and a cavalry elite.

More important and relevant to the question though: if you are raising the question of why European armies didn't adopt the "successful" warfare of the steppe nomads, remember that European armies were also successful in the wars they fought against each other and, in this time period, against Muslim kingdoms. Even with the Mongol invasions, the Mongols did not penetrate that far into Europe, where their method of warfare might have been less effective in European geography. Similarly, the Huns were able to amass terrifying armies against a disunited Europe, but they too were soundly stopped in pitched battle at the Cataluanian Fields.

If European warfare was unsuccessful, it would have evolved to adopt more effective methods. This is best seen in the Roman Empire in the aftermath of the Parthian Wars, and later armies of the Roman Empire adopted the same horse archers and armoured cataphracts to be used in warfare in the Eastern regions. Warfare adapted to suit the region - and similar, the same could be said of South Asia, with its mix of deserts, plains and jungles, where the culture of mounted warfare was not integral to the pastoral kingdoms and empires of the region.