Eastern Christianity is a very diverse facet of the Christian faith. For example there are Eastern Catholics ( Churches that are in communion with the Bishop of Rome of which the Maronite Catholic Church is the oldest, having never separated from the Catholic Church at all.) There are also, the Orthodox family of Churches of which there are the Oriental Orthodox Church (which consists of the Syriac, Coptic, Indian , Armenian and Ethiopian Churches) and the Eastern Orthodox Churches (which include The Greek, Russian, Serbian, Romanian and other Eastern European Churches) These two groups of Orthodox Churches differ mainly in terms of their Christology. The Oriental Orthodox should be rightly called Non-Chalcedonian Miaphysite Orthodox Churches since they reject the Christological formula defined by the Ecumenical Council of Chalcedon that posited that Christ had in His One Person (Hypostasis in Greek) Two distinct Natures (Physis in Greek) , One Human Nature and One Divine Nature, and instead subscribe to the Christological formula proposed by the Alexandrian school of theology that maintains that in the One Person of Christ, there is One Nature that is both Human and Divine and without a separation of His Humanity and Divinity hence the term Miaphysite which means One Nature.
Eastern Orthodox Churches however accept the Christology of the Council of Chalcedon and hence should rightly be called Chalcedonian Orthodox Churches and therein lies the difference. The similarities between the the two Churches are that they are autocephalous (self-headed or self-legislated), that they dispute the nature of the authority of the Pope of Rome and argue that he is “first among equals” or primus inter pares in Latin. Another similarity is that all Orthodox Christians Chalcedonian and Non-Chalcedonian, contend that the addition of a phrase to the Nicene Creed that says that the Holy Spirit proceeds from the Father and the Son (Otherwise known as the Filioque in Latin which Catholics affirm) instead of the Father only as indicated in the first edition of the Nicene Creed is erroneous and invalid.
There is also another Church known as the Assyrian Church of The East that is neither Orthodox nor Catholic (in communion with the Church of Rome). The Assyrian Church of The East is characterized by how they understand the Archbishop Nestorius whom they regard as a saint, They contend that Nestorius was wrongly condemned for his Christological and Mariological views and while they do not subscribe to certain aspects of the theology of Nestorius, they do still use the term that Nestorius used as a title for the Virgin Mary which is Christotokos (Mother of Christ or Christ-Bearer) instead of the term used by Catholic and Orthodox Christians, Theotokos (Mother of God or God-Bearer) and also maintain like Chalcedonian Orthodox and Catholic Christians that Christ has Two distinct Natures that are both Human and Divine. Also, The Assyrian Church of The East affirms the position of it’s Orthodox bretheren in not accepting the addition of the Filioque to the Nicene Creed with regards to the procession of the Holy Spirit
Note: This is but a brief description of Eastern Christianity, for more information on the topic do consult the Blackwell Dictionary of Eastern Christianity that is a truly comprehensive resource on the subject. Cheers!