Areas of agreement and differences between Eastern Orthodox Christians and Roman Catholic Christians

Back in 451 AD, the Egyptian Church split with the Roman/Greek Church. The split came after the Chalcedon council created a new patriarchy in Constantinople with greater authority than the existing one in Alexandria, Egypt. I suspect that change was far more important in the split than theological fine points.
I wouldn't be surprised if there was a lost subtext of Church politics and jockeying for power. However, I recommend reading more about the Chalcedonian split before putting it in those terms. This conflict could be traced back further than Chalcedon, even back to Nestorius and Cyril (of Egypt) and the Second Ecumenical Council.
- Nestorius theorized that Christ had two hypostases, and Cyril denied this, saying that two hypostases would entail that Christ had two persons. Nestorius denied that his theory entailed that Christ had two persons. I am open-minded on whether Cyril interpreted Nestorius correctly or not, since two hypostases could be translated as either two substances or two subsistences, and the Churches have agreed that Christ has two substances. But in any case the 2nd Ecumenical Council decided that Cyril was correct and that Christ had only two hypostases.
- The conflict led over into Cyril debating with the Antiochian school over whether Christ had two natures or just one. The Antiochians said Christ had two, Cyril asserted that Christ had one (miaphysia), and eventually Cyril said that he (Cyril) and the Antiochians were really in agreement, and he smoothed relations over. Problem averted? Unfortunately not, because...
- ... Cyril's successors in Egypt asserted that it was a heresy to teach that Christ had two natures. Meanwhile, the Patriach of Constantinople deposed Eutyches for claiming, among other things, that Christ didn't have two natures. Eutyches' statements included: Christ had two natures before the union, but only one afterward.
- Pat. DIoscorus in Egypt supported Eutyches, called another council in Ephesus and reinstated Eutyches and excommunicated the Patriarch of Constantinople.
- The EO and RC side called Chalcedon to teach that Christ had two natures.
Yes, church governance has always been a driving issue.
Politics matter in real life.
Probably in real life terms, the imperial power of Rome played a key role in building up the power claims of the Vatican. Before then, the Eastern Roman empire was comparable in political power, and so it would be harder politically for the Vatican to press or puff up its claims.
You may be interested in this book about the period from 450 to 680 AD:

I found the history of this period to be interesting because in many ways it is the reverse of recent history. Here are some examples:

1) The western Church was fragmented into regional churches that operated more-or-less independently of Rome. In many parts of western Europe had Germanic rulers who followed Arian Christianity, but the population was mainly Catholic. Remote areas such as Ireland had little contact with Rome.

2) On the other hand, the areas of the eastern Church were under the control of the emperor in Constantinople. The emperor also ruled over most of Italy including Rome.

3) The popes were elected by the people of Rome, but the emperor approved the selection. Most other bishops were not subject to approval of the emperor. Several of the popes spoke Greek as their native language.
You may be interested in this book about the period from 450 to 680 AD:

I found the history of this period to be interesting because in many ways it is the reverse of recent history. Here are some examples:

1) The western Church was fragmented into regional churches that operated more-or-less independently of Rome.

2) On the other hand, the areas of the eastern Church were under the control of the emperor in Constantinople.
It's true that in Western Europe after the fall of Rome to the Goths in about the 5th century, political power was more regional, whereas the eastern emperor retained control of the eastern meditteranean until the rise of Islam (7th century) and the conquest of Constantinople (15th century). But in contrast, in the west, it's considered that Church power was more centralized under the Pope than Church power was with Constantinople- in the East there were 4 Patriarchates. Still, one would want a theory of how it happened that Rome ended up claiming such strong supremacy claims for itself, claims that grew over time, to where in the 19th or 20th century, the Pope began asserting Papal Ex Cathedra Infallibility dogmatically, which his office hadn't asserted in medieval times.
What do Eastern Orthodox and Roman Catholics agree on?
I was born and raised in the RCC so I have a slightly different perspective which I'd like to illustrate via comment on your OP.
  • The first 7 Ecumenical Councils
  • Christ's divine and human nature
  • Apostolic succession
They lost that the moment the bull of ex-communication was slapped onto the altar of the Hagia Sophia in Constantinople, and likely even a few hundred years before that. Now they base their authority almost entirely on Peter being their first Pope.
  • Bishops, presbyters, deacons and their ministries
they add Cardinals, Jesuits, Benedictines, women altar servers, etc...
  • The broad structure of the visible Church
Absolutely destroyed by the addition of the Vatican, its Cardinals, and its Pope as having the ability to make infallible declarations.
  • Invocation of the saints
None of whom they venerate seeing as the icons have all but vanished.
  • Honor due to the Blessed Virgin Mary as Theotokos
"Born sinless" and "assumed bodily into heaven before death", both foreign to Apostolic tradition.
  • Acceptance of the 7 sacraments (although the Orthodox doesn't put a limit on the number of sacramnets)
..each one skewered from the Apostolic tradition (except maybe ordination and anointing of the sick). Confession isn't "absolution at the pronouncement of a priest hidden in obscurity who declares legalistic forms of penance", Baptism is via sprinkling or pouring (generally), Chrismation is delayed approximately 12 years, Holy Eucharist is delayed approximately 8 years, and Marriage might as well be Protestant in form and function.
  • Confession in the presence of a presbyter
hidden behind a wall who declares absolution and a legalistic form of penance as opposed to suggestions to help spiritual growth and the shedding of sin
  • Use of icons in worship
..statues and stations of the cross aren't "icons".. there are very few to no icons in any given RC church and, even if present, certainly never get venerated or treated as anything more than a painting -- BIG mystery to me how and when they disappeared in the first place, but Satan has been busy ensuring that RCCers are not visually reminded of their eternal brethren in the context of Holy Scripture.
  • Celebration of the Eucharist and affirmation of its sacrificial nature as identical with the sacrifice of Christ
delayed for about 8 years for little children (who were incorrectly baptized -- see above) and won't receive "Confirmation" for another 4 years, the blood of Christ was held back from the laity for a number of years in many places, uses unleavened bread (WE are to be unleavened via Confession and our liturgical and personal prayers for forgiveness and thereafter receive the GOOD leaven of Christ's body in Holy Eucharist)
  • The Eucharistic bread and wine becoming the Body and Blood of Jesus Christ
the PRE-crucifixion body and blood of Christ, not the RISEN & GLORIFIED Body & Blood of Christ -- hence stuff like their doctrine of transubstantiation (as opposed to our mystery of transformation) and stigmata
I like this analogy which I read about when speaking of the differences between Orthodox and Roman Catholics. It goes like this, "Imagine for a moment: an American travels to Europe and sees a European there. He looks at the European, and after assessing him for a moment, says to himself, 'We are so similar'. But the European looks back at the American, studying him deeply, and thinks, 'He could not be more different from me'. In this scenario, the European is the Eastern Orthodox and the American the Roman Catholic. What are we trying to say here? The differences between our churches are far more numerous from an Orthodox perspective than they are from a Catholic perspective.
Agreed, and well put
Many Catholics believe our churches are closer in beliefs and practice. To an extent, this is true, because we certainly share more in common with one another than we do with the Protestant world. However, there are several differences between our churches, and those inquiring into either of them deserve to know what those differences are."

Differences between Eastern Orthodox and Roman Catholic Christians:
  • Development of doctrine
    • The catechism of the Catholic Church states in paragraph 94 that the Holy Spirit causes, "the understanding of both the realities and the words of the heritage of the faith is able to grow in the life of the Church.” As Orthodox we would see the Immaculate Conception or Papal infallibility as developments of doctrine.
    • The Orthodox Church practices the development of the expression of doctrine. When the early Church declared a dogma, it's purpose was to respond to a heresy at the time. "Orthodox dogma never claims to expound the whole truth about anything. Instead, it only delineates the borders of the mystery, which God Himself revealed to us in the way He chose to reveal it."
  • Liturgical reform
    • I am not saying that the Orthodox Church has not has not gone through changes in the Liturgy. It has, which can be seen in the liturgical manuscripts. I am also not saying that Vatican II is the reason for bad Liturgy in some Catholic Churches. However, as Orthodox Christians, if we were in communion with the Roman Catholic Church, we would want to go into a Roman Catholic Church, celebrate Mass, and feel at home. I have been to a NO Mass that stuck to the rubrics, which was a sung Mass with Gregorian chant, the priest facing the same direction as the people and the use of incense. But unfortunately, this is few and far between.
I didn't realize that existed outside of their Uniate churches (Eastern Catholics).
  • Papal Supremacy (the BIG one)
    • The Orthodox do not hold to the ideas that the Pope of Rome has immediate, supreme and universal jurisdiction over the entire Church.
not to mention the College of Cardinals
  • Papal infallibility
    • This dogma of the Roman Catholic Church teaches that the Pope of Rome, when teaching on questions that have to do with faith and morals, due to his office, cannot err.
only doctrines and theology that have passed through the entire body of Christ in between Ecumenical Councils (and finally certified by the next Ecumenical Council) are declared as Holy Tradition of the church
  • The filioque
    • This added phrase to the Nicene Creed. Speaking about the Holy Spirit, "who proceeds from the Father and the Son (filioque).
that has caused more confusion and division over the centuries
  • Absolute divine simplicity
Please explain. New one on me.
  • Created Grace
Please explain.
  • The Immaculate Conception
Mary was not only "born without 'original sin'", but was raised bodily into heaven before her physical death. I think the whole concept of "original sin" started their doctrines about Mary
  • Purgatory
    • Orthodox do believe in purification of the soul, however, the Orthodox have not defined it any further.
I would like to find a good book on this subject. My current viewpoint is that we're cleansed but not in a legalistic sense.
Something for non-Orthodox and non-Catholic Christians to chew on. I know that many see the Orthodox as Roman Catholics without the Pope.
Orthodox is the RCC without centuries of heresies and errors piling up to change the church so much as to make it unrecognizable to the Apostles.