The Eastern tradition likens yeast in bread to the soul in the body. The soul gives life, and therefore the "living bread" of the Eucharist must have yeast. The West uses unleavened bread because that is what Jesus used in the Last Supper.
When the Orthodox Church broke from Rome in 1054, the Patriarch of Constantinople condemned the West for using unleavened bread, but that was a spurious charge. The Council of Florence approved the use of either kind of bread in 1439, so the use of leavened or unleavened bread is a question of licitness, not validity. This was infallibly defined.
The council stated, "We have likewise defined that the body of Christ is truly effected in unleavened or leavened wheaten bread and that priests ought to effect the body of our Lord in either one of these, and each one namely according to the custom of his Church, whether that of the West or of the East" (Decree for the Greeks).
Roman-rite Catholics are not permitted to use leavened bread, however (Code of Canon Law 926). The Church desires uniformity to show that the sacrifice of the Mass is the same sacrifice everywhere. Using leavened bread would not invalidate the Eucharist, but it would be a grave error to disobey the Church and a long, venerable tradition. Eastern-rite churches in communion with Rome are allowed to retain their own tradition of using leavened bread.