The importance of the armed cavalryman in battle transferred to his social status outside of battle. Knights ranked higher than foot soldiers. Part of this was the cost of outfitting a mounted warrior: few could afford it, which made knights not only special for their ability, but also because of their rarity (compared to infantry).
This special significance in warfare ultimately faded, especially once the English longbow men proved to be so valuable and deadly during the Hundred Years War, such as at Agincourt. The cavalry evolved into a way to get fighters to the battlefield fast, who then dismounted and used swords, maces, and poleaxes to fight on foot, engaging the enemy on its own level.
It occurs to me that the Hundred Years War has been mentioned many times, for instance here, but never explained. I'll give it a crack tomorrow.