The detail
The Church of the East grew up beyond the Roman frontier, in Mesopotamia and Persia, using Syriac as its language of worship and theology. Because Persia was Rome's chief rival, Christians there were often suspected of loyalty to the Roman emperor and suffered periodic persecution. Partly for that reason the church organised itself on its own: the Synod of Seleucia-Ctesiphon (410), under Catholicos Isaac, accepted the Nicene faith and set up a hierarchy headed by the catholicos of the Persian capital, and the Synod of Markabta under Dadisho (424) declared that the catholicos could not be judged by, or appealed against to, the "Western Fathers", that is, the bishops of the Roman Empire.
The theological break came with the Christological controversy between Nestorius of Constantinople and Cyril of Alexandria. The Council of Ephesus (431) deposed Nestorius and approved calling Mary Theotokos ("God-bearer"). The Persian church took no part in the council and never received it. Its theological tradition looked to the Antiochene school, above all Theodore of Mopsuestia (d. 428), revered in the East as "the Interpreter", whose writings were studied at the School of Edessa. When Emperor Zeno closed that school in 489, its teachers moved to Nisibis in Persian territory. The Synod of Beth Lapat (484), led by Barsauma of Nisibis, and the Synod of Seleucia-Ctesiphon under Catholicos Acacius (486) committed the church to a two-nature (dyophysite) Christology in the Antiochene line; the synod of 486 also allowed clergy to marry.
The church's mature Christology was set out by Babai the Great (d. 628): in Christ there are two natures (kyane) and two qnome, a Syriac term roughly meaning concrete individual realities, united without mixture in one prosopon (person or presentation) of the Son. Its opponents read this as dividing Christ into two subjects, which is what "Nestorianism" means in Greek and Latin polemic. The Church of the East has always rejected that charge, and many modern scholars (notably Sebastian Brock) call the label "Nestorian Church" a misleading misnomer: Nestorius is honoured as a Greek doctor of the church, but he is not its founder and played little role in shaping its theology.
Politically the separation suited both empires: the Persian kings could tolerate a church that was visibly independent of Constantinople. From its base in Mesopotamia the Church of the East became one of the great missionary churches of the first millennium, reaching Central Asia, India (the St Thomas Christians) and China, where a stele erected in 781 records its arrival in 635. Its later history includes heavy losses under the Mongol successor states and Timur (late 14th century), a Rome-linked branch beginning with Yohannan Sulaqa (1552–1553) that developed into the Chaldean Catholic Church, and a 20th-century split (1964–1968) between the Assyrian Church of the East and the Ancient Church of the East.
How each side tells it: in the Byzantine, Latin and Oriental Orthodox narratives, the Church of the East is the community that held to the error condemned at Ephesus. In the Church of the East's own narrative, it simply kept the apostolic faith of the East and the teaching of the Antiochene Fathers, and was never party to a council it was not invited to. Modern dialogue has narrowed the gap: in the Common Christological Declaration (1994), Pope John Paul II and Catholicos-Patriarch Mar Dinkha IV affirmed that they confess the same faith in Christ despite different formulas, and in 2001 the Catholic Church recognised the validity of the Anaphora of Addai and Mari, an ancient Eucharistic prayer that lacks a narrative of institution.