Monks, not Priests
The Christian church in Europe developed a strong so-called "lay" hierarchy, meaning the ladder of Pope, bishops, and parish priests; parish churches and cathedrals formed the core network; the monastic orders came late, and while rich and powerful, always played the second fiddle.
But with Buddhism, the monks and monastic communities are the foundation; technically, there is no church structure other than monastic structure. Places of worship for laymen are an afterthought, and are always attached to monasteries, physically or at least administratively.
So, religion in medieval Buddhist Europe would be focused on a loose network of monasteries. Abbots, not bishops, would be powerful; and there would be no Pope and no formal top-down hierarchy. The older monasteries would be more influential, especially with regard to "daughter" monasteries they founded, but there would be no formal chain of command that the Christian church so insisted on.
On the other hand, the Empire would thrive. In Catholic Europe, it was mostly the Church that, quite deliberately, broke the power and influence of the Emperor in order to strengthen the Papacy; this resulted in the breakup of the Empire into a multitude of competing sovereign nation-states. But the Buddhist structure would see no advantage in weakening the Empire. On the contrary, they would benefit from a strong central administration that would protect and subsidize the peaceful monasteries. This is the pattern from all the Buddhist countries; the monks supported Ashoka, the Son of Heaven, and the Tenno, regardless of the fact that, technically, only Ashoka was an actual Buddhist.
So, in a Buddhist Europe, there would be no France, England, or Spain; or more precisely, they would exist as more-or-less autonomous provinces of the Holy Roman Empire. This could result in a very different dynamic. Local warfare would be reduced, due to both Imperial enforcement and Buddhist peace ideals. But the advance of Islam, rather than encounter loose resistance by local Spanish and French forces, would be faced with the professional army of a continent-wide united Empire, rather than just the Byzantines. The advance would likely be stemmed in mid-Africa. On the other hand, the retributive Crusades would not occur, especially considering that Jerusalem has no religious significance to the Buddhist faith.
Italy would not have the historical importance that it had in RL; there would be no Holy See with its immense political clout, and the merchant republics of northern Italy would not exist as independent city-states. The Renaissance, if it happened at all, would occur somewhere else. Buddhist missionaries would reach into Russia, and into Africa. Trade over the Silk Road would expand. The discovery of the Americas would probably be long delayed, for two reasons: firstly, there would be no significant Atlantic fishery, as Buddhists do not consider fish a Lenten staple; secondly, there would be no need to establish alternate access to the Orient. It is anyone's guess how the Mongol invasion would develop, given well-established relations between the European and Chinese empires. With a centralized army, and without constant internecine conflict, military technology would probably take much longer to develop to the gunpowder stage, and the Industrial Revolution might not happen at all, being replaced by slow, gradual development, with frequent setbacks, following the Chinese model.