What is the difference between the Roman Catholic and Greek Orthodox churches? What is the difference between the Roman Catholic and Greek Orthodox churches?
The differences are almost too great to list in an answer like this! The real problem in answering your question is that it isn't just a list of "things Roman Catholics believe" and "things Greek Orthodox believe". (NB that "Greek" Orthodox probably isn't accurate here: it's more accurate to talk in terms of "Eastern" Orthodox.) The problems are much more fundamental and enormous than that, and ultimately amount to entirely different methods of theology, theological reasoning, ecclesiological assumptions, thinkings about authority, worship, etc. A major factor is the simple fact that the Western Church (which became the Roman Catholic Church) spoke and wrote in Latin, whereas the Eastern Church used Greek. Translation (particularly of theology) is hard. 

But here are some headlines:

Authority The Roman Catholic Church believes in the infallible teaching authority of the Church. This most famously means the doctrine of papal infallibility, though the concept of the infallibility of the ordinary magisterium (teaching office of the church) is also stated. This is a concept foreign to the Eastern Orthodox, who see the authority of the Church as stemming from the Holy Tradition, seen most fully in the Ecumenical Councils. But the Orthodox would be careful not to pin the question of "infallibility" down: it's not a concept that really works in Eastern thought.

Papacy For the Roman Catholic Church, the Bishop of Rome, the Pope, has universal jurisdiction over all the Church. This is not true in Eastern Orthodoxy. They recognise the Bishop of Rome as being primus inter pares, first among equals, but as essentially equal to the other major patriarchs (Constantinople, Antioch, Alexandria, Jerusalem).

The filioque clause The famous cause of the Great Schism of 1054. This was a modification to the Niceno-Constantinopolitan Creed. The question was whether the Holy Spirit proceeds from the Father, or from the Father and the Son. (filioque means "and from the Son".) For the Catholics, it is important that the phrase is used; for the Orthodox, it is breaking with tradition and perverting the doctrine of the Trinity. This problem is closely related to the above problems.

Church unity What is necessary for the Church to be one? For Roman Catholics, it means participation in the organisation headed by the Pope. For the Orthodox, it means membership in one of the Orthodox churches, which are fully in communion with one another.

These are all "dividing" issues, which contribute to the continuing division between the Orthodox and the Catholics. There are other differences as well, which aren't divisive in the same way:


Liturgy The Roman Catholic Mass is very different to the Orthodox Divine Liturgy.
Married clergy In the Roman Catholic Church, priests may not be married (except in some narrow circumstances, and in some Eastern Catholic Churches). For the Eastern Orthodox, married men may become priests, but not bishops.
Bread and the Eucharist Roman Catholics typically use unleavened bread (wafers) whereas the Orthodox use leavened bread. This is due to a very old dispute about the precise date of Easter, and whether the Last Supper was a Passover meal.


And, moreover, culturally there are very different attitudes. A summary would be that Catholics tend to systematise things (e.g. the Summa Theologica of Thomas Aquinas) whereas the Orthodox tend to prefer a more discursive, poetic, less precise way of doing theology.