What importance did the prisoner of the church union attach to Western Europe and Byzantium?

For Western Europe, Union primarily meant that Orthodoxy (or the orthodox faith) of Byzantium would become dependent on the Pope. And thus, over time, the split of the church into two parts would still be excluded. And the Pope himself became the most influential person, received additional lands for influence.
Byzantium hoped for help from Western Europe against the Ottomans. Including real military aid. However, the European states were not eager to die in distant Byzantium.