The Alexandrian School
Emphasized the unity of Christ’s person as divine. This center of early Christian Platonism reflected a tendency to assume history into eternity, matter into spirit, historical exegesis into allegorical (spiritualizing) exegesis, and the humanity of Christ into His deity. Its celebrated representatives include Origen, Clement, and Cyril. Though this school was formally orthodox (with the exception of Origen), its excesses led to Apollinarianism and Monophysitism (Eutychianism), which were judged heretical at the Councils of Constantinople (381) and Chalcedon (451).
The Antiochene School
Emphasized the distinction of Christ’s person as divine and human, favoring the human. Sharply critical of the allegorizing hermeneutic of the Alexandrians, this school closely followed a historical-literal hermeneutic. Among orthodox theologians, this school produced Diodorus of Tarsus (died in 390), Chrysostom (347-407), and Theodore of Mopsuestia (350-428). The excess of the Antiochene emphasis emerged first in Theodore – who refused to acknowledge Mary as “Theotokos” (God-bearer) – but especially in his student, Nestorius. Nestorianism was also condemned by the councils mentioned above.
From: The Christian Faith: A Systematic Theology for Pilgrims on the Way by Michael Horton (Grand Rapids: Zondervan, 2011), p. 473.