Why is there a Theological difference here if both had the same faith?
The Eastern Churches do not have the “same” faith as the West. They have the same origins, but the Church of the East has chosen to reject all formal teaching since the Schism; thus their faith is stagnant and less complete.
The West enjoys the continuing teaching Grace of the Holy Spirit as exercised in the Universal and Ordinary Magesterium, as well as the Extraordinary teachings of Popes and Ecumenical Councils. The East has turned away from these teachings.
The Bible offers us some understanding of the Fall, but it does not address some further questions. Neither does the limited body of doctrine considered authoritative by the East. So Eastern Christians must turn to the Early Church Fathers for guidance which is not considered (by them) to be definitive, but is nevertheless substantial. Lacking in any ability to define doctrine themselves, Eastern Christians tend to regard ECF’s more substantially than Western Catholics.
However, whereas Western Catholics hold the teachings of ALL Early Church Fathers in high esteem, Eastern Christians tend to regard only the writings of Eastern ECFs. Thus, they have little regard for St. Augustine of Hippo (who attested to the same views as presently held in the West). Eastern Christians defer to St. John Cassian, who held a more moderate view (which would later come to be known as semi-Pelagianism).
It depends on what you mean by “Apostolic.”
We have no historic documentation that any Apostle taught about the Fall in any more detail than the Bible itself. So, in this regard, neither understanding is “Apostolic”
However, if you mean which teaching is endorsed by the “One, Holy, Catholic, and Apostolic Church” then the answer should be obvious, as there is only “One” Church that has valid claim to every one of these four notes.