Well, we may presume the battle between the Papacy and the Hohenstaufen to go on but more overt. In otl, at the Council of Lyon, the Papacy iterated its conflict with Frederick II (1220-1250 CE) but this iteration was shadowed by the proclamation of a crusade against the Mongols, which inadvertently assisted his rival in Germany who was able to more seriously acquire resources and allies for which to combat and frighten the Mongol incursions from Hungary.
The Papacy nevertheless proved itself able to break Frederick II despite the Mongol fears to the north and east. Papal powers exerted overt crusades, possibly intended for the Mongols upon the Holy Roman Emperor alongside a collection of Imperial allies of the Papacy against Hohenstaufen hegemony. Ultimately, the Papacy broke the Hohenstaufen power in Italy and in the Empire and asserted a Papal supremacy in the region that had been lessened after the death of Innocent III.
However, without the Mongol menace, we may see less peasant sensationalism in Europe pertaining to a great invasion from the east and the necessity to gather a crusade. Up to that point, sensationalism amongst the peasantry for war and clamoring for a Crusade, was reasoned as a reclamation of lands once held and protection of pilgrims. However, an apocalyptic angle related to the fall of Europe, and the end of the world may not exist. Thus giving traveling troubadours less sensational rhetoric in atl.
