Dear Incogneat,
An excellent question, sir!
In fact, the devotion to the Sacred Hearts was and is practiced by many Byzantine Catholics - it was most popular among our New Martyrs and Confessors of the Soviet Yoke, such as Bl. Paul Gojdicz and others. Patriarch Joseph the Confessor was especially devoted to the Sacred Hearts and insisted on placing prayers to them in his prayer book.
In addition, devotion to the Heart of Christ IS practiced in Orthodoxy, but not in the way it is in the West.
In the East, there are references to the Wounded Side of Christ in the Octoechos, the liturgical collection of hymns and canons for the variable portions of the Divine Liturgy and the Hours.
Orthodox Saints in BOTH the Kyivan and Greek traditions venerated the Heart of Christ. St Dmitri of Rostov was such a one and St Nicholas Cabasilas was another (in fact, ST Vladimir's Seminary, in publishing a Foreward to Cabasilas' work on the Divine Liturgy HAD to admit the Saint's references to the Head and Heart of Christ - but also added that this was different from the devotion to the Sacred Heart of Jesus in the West).
Orthodox often kiss the edge of the Chalice after receiving Holy Communion to venerate the Wounded Side of Christ as well . . .
As for differences, these are:
1) The East does NOT represent the Heart of Christ upon His Chest, as is done in the East - the East is against what some have called the "worship of parts of the Body of Christ" As I understand the Western theology guiding the devotion to the Sacred Heart of Jesus, it is also forbidden to represent the Heart apart from the rest of the Body of Christ.
2) The East, instead, venerates the entire Christ as the "Lover of Mankind" with icons of Him holding a Gospel text (sometimes actually decorated with heart symbols) with words referring to His Love e.g. "Greater love has no man . . ." or "Another commandment I give to you, that you love one another, even as I have loved you" or "Come to Me all you who are weary and I will give you rest etc." Sometimes there are icons that also indicate Christ pointing to His Wounded Side, as in the Greek Icon of "Ho Eleimon" or "Christ the Merciful" that I have.
3) In prayers to Christ the Lover of Mankind, the East would NEVER address His Heart as though a Person, as occurs in the West. I've read RC devotional texts that say contemporary RC's tend to shy away from this devotion and pray to Jesus directly, mentioning His Heart etc.
4) There are several icon types that focus on the heart/soul of the Mother of God. One of these is the famous "Seven Arrows" referring to the arrows or swords that Simeon prophesied would pierce the soul/heart of Mary (in the Octoechos I have, the words "soul" and "heart" are often used interchangeably). This depicts the Mother of God with seven arrows in various positions buried in the centre of her chest - at no time is her heart ever depicted. Dave Chtec had an avatar with that icon but he keeps changing his avatar like some people change their socks!

Also, there is the beautiful icon of St Seraphim of Sarov, the "Joy of all Joys" that is a depiction of the Theotokos at the moment of the Annunciation with hands crossed over her chest and her eyes looking down into her heart.
5) The Hesychastic Jesus Prayer spirituality also focuses on the heart as the "prayer of the heart" and it is no accident that saints like St Seraphim prayed before the icon of the "Joy of all Joys" as a way to focus on this heart spirituality - at no time are depictions of the physical heart actually employed.
Alex