It is my understanding that at the siege of Constantinople in 1453 there was a contingent of Venetians among the Byzantine defenders. What changed in the relationship between Venice and Byzantium in the years between the Venetians sacked Constantinople and when they defended it, if anything changed? Were the tensions between the Western and Eastern church lessened? Was it just because the invaders were Muslim?
Essentially, one of the prime reasons for the Fourth Crusade targeting Constantinople was that the Venetians wanted to oust the Genoans from the city and claim the incredibly lucrative trade rights for themselves. It stands to reason, therefore, that they'd then have a stake in defending their trade quarter.
E: Sources: Epstein, S. (2001), Genoa and the Genoese
Queller, D.E. and Madden, T.F. (1997), The Fourth Crusade: The Conquest of Constantinople:
Genoa had been losing a trade war with Venice for control of the Balkans and Dalmatian coast, but in return had been negotiating with the Byzantine Emperor to exclude the Venetians from Constantinople and establish trade dominance there. Given that the Venetians were supplying the transport for the Fourth Crusade, it's highly likely they saw this as an opportunity to 'correct' this and essentially scupper Genoese mercantilism for some time, which is indeed what happened. The French contingent of the crusade was already distrustful of the Genoese for "letting them down" on the Third Crusade in terms of transport.
The Pope's control over the Crusade waned rapidly, especially after its leaders signed the Treaty of Venice in 1201, which essentially made them a Venetian mercenary army. Indeed the terms granted a half of all gains from the Crusade to Venice, in payment for the fleet that would transport them. They originally planned to attack Alexandria but the Crusader States had signed a 5-year truce with Egypt in 1198 and had no interest in resuming hostilities. The Byzantine Emperor was relatively new, however, and had made enemies of his brother and nephew, who travelled to Europe, and Venice in particular, offering land, riches, power and trade in return for delivering the Imperial throne to them. This coincided very neatly with Venice's needs. Eventually the Crusade agreed to target Constantinople, under the flimsy justification of bringing them back to the Latin Church.
Venice ended up with a trade quarter in the city and it remained an important outpost of the Venetian trading empire right up until the fall of the city in 1453, a period during which Venice was also occasionally at war with the Ottomans.