How did the Crusades affect the relationships between the Eastern and Western Churches?
The Crusades affected the relationship between the Eastern and Western Christian denominations in several ways. The Byzantine Empire based around Constantinople had been weakened in the centuries and decades due to conflicts with the Turks and Muslim armies that would come to occupy the Holy Land, so the Emperor, Alexios I Komnenos called upon Western Europe and the current Pope Urban II for aid.
An important early point as to what the relationship between East and West was to be despite the shared goal was the desire of Alexis I for all of the crusading nobles (essentially the leaders) to swear an oath that all captured lands would fall under Byzantine control. It wasn't long until some nobles, such as Tancred, balked at this demand, given that many Crusaders sought land and wealth well before glory for God in the Holy Land. Thus, a contentious relationship was established right off the bat...
This was exacerbated by another reality of the type of soldier who went on Crusade. While some were doubtlessly devoted to the cause of liberation, the fact remains that crusading was a common solution for second and third noble sons and suchlike who were distrusting the status quo in Europe. The promise of wealth was a powerful motivator, and remissions of sins before, during, and after was considerable as well. This created a recipe for many who were unconcerned about diplomatic relations to essentially wreck havoc while crusading, doubly so because anyone who didn't look like them was seen as the enemy. Either willfully or not, the results were quite brutal, and certainly didn't serve to paint the best picture of supposed European civility to the Byzantines and other citizens of the Levant.
These realities persisted as the Crusades did in success and failure until the tragedy of the Fourth Crusade, in which a failure based on dynastic quarrels escalated into a sacking of the city of Constantinople, quite off track from the original goal of attacking Egypt on the way to another attempt at the Holy Land. This brutal act that lead to the short lived Latin Empire and centuries of deserved shame for the Western Catholic Church also served to line the pockets of Venetian merchants who originally put up the money for travel to Jerusalem. These profiteers ended up with many of the city's greatest treasures (the Horses of Saint Mark) leading many historians to question if the sack of Constantinople was the goal of the Fourth Crusade from the very start.
Even though reconciliation over this crime against humanity was eventually achieved, there is a reason that it was the culmination of the Great Schism between the two churches and the most significant event that contributed to the decline of the Byzantine Empire and its eventual conquest by Mehmed the Conquerer and the Ottoman Turks.