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.
I would like to recommend what lonesomeday wrote, but with a few minor adjustments.
Authority Part of the reason Orthodox do not like to pin down the question of infallibility is because it is understood among them that we live in a broken, sinful world. But the question of infallibility does not concern the Pope of Rome alone. A certain type of infallibility and incorruptibility is placed on the scriptures by many modern Christians. The Orthodox have maintained a stance on the scriptures that continues the thought of the early church on them. While the scriptures were seen as the infallible word of God expressed through fallible men, it was believed that, were they all to be burned, the Holy Spirit would simply reveal them again, even if they appeared with different wording. (This thought has been expressed in various ways by many different authors over the centuries.) This idea sheds light on just how foreign the idea of infallibility is to eastern sensibilities. Language itself is viewed as part of the corrupt experience of mankind. We must grapple with it in trying to express ourselves. The very fact that numerous interpretations have been made of the same words bears out the wisdom of this understanding. This is also part of the reason why councils are so important to Orthodox. Language is used and its interpretation is agreed upon. So, as lonesomeday suggested, it's not simply that Orthodox reject the infallibility of the Pope: infallibility is completely foreign to Orthodox thought.
Papacy The Bishop of Rome was seen not simply as first among equals and "essentially equal to the other major patriarchs"; rather, an equality in terms of vote existed at all councils. Each bishop had one vote (except in cases where one bishop was voting on behalf of another). Even so-called "country bishops" had the same voting power as a patriarch. Then, what is a patriarch or pope? The term patriarch refers to pockets of centrality of administration, not centrality of spiritual power. From a practical point of view, bishops holding sees with a great flux of people and thought were in a better position to distribute and hear teaching, tradition, and practice. As such, patriarchs presided at their respective local councils, but they still had the same voting power as the last bishop to vote.
The term pope is one of endearment. It was first applied not to a bishop of Rome but rather to the thirteenth bishop of Alexandria, Heraklas (232-248) in a letter from his successor, Dionysios, to Philemon. About three centuries later, it was used by the bishop of Rome, John (523-526). The term is obviously still in use for Rome, but many do not realize that to this day, the bishop of Alexandria is referred to as "The Pope and Patriarch of Alexandria."
But as to the current meaning of pope among western Christians, there is a great difference between Orthodox and Roman Catholic thought. Strangely, the Orthodox thought on this is best expressed by St. Gregory the Dialogist (or St. Gregory the Great), who was a bishop of Rome (540-604). During his time, a controversy arose over the term ecumenical. Many clerics and courtesans urged the then patriarch of Constantinople, John, to begin using the term ecumenical (oikoumenikos). The term has two distinct meanings, depending on its usage. John was urged to use it as meaning "imperial." The imperial patriarch made sense, since Constantinople was, after all, the imperial capital.
But the other meaning of ecumenical is universal, and this is how it was understood by Gregory. He immediately wrote to several other bishops denouncing the title. He described Christ as the head of the church and decried the idea of one bishop being universal, or one bishop being greater than another. His language is quite strong on this: "Who is he who calls himself a bishop of bishops? He is an antichrist." It is also interesting that he appealed to other bishops to help resolve the dispute. Like St. Gregory, Orthodox reject the idea of a bishop of bishops.
The filioque Clause The phrase was first used in the west in an attempt to fight particular heresies. The objection to its use in the East was not to its application in the temporal procession of the Holy Spirit (i.e., during the life of Christ) but rather to its application in the eternal procession. All this probably sounds like splitting hairs, but it actually makes the Holy Spirit a creature from the eastern perspective. The most thorough argument in this regard was given by Photios the Great in his reply to Pope Nicolas, entitled, "The Mystagogy of the Holy Spirit." (Forewarning to anyone attempting to read it: its incredibly difficult language frustrated the scholars of the papal court immensely, so don't be too freaked out if you can't comprehend what it's even trying to say on the first attempt.) The Roman church has at various times recanted its use of the filioque. Pope Leo was so against its use that he actually had silver plaques made to be placed in St. Peter's Cathedral without the filioque in the creed. More recently, the just retired Pope Benedict XVI declared that the creed containing the filioque would no longer be taught to catechumens.
Liturgy One interesting difference is the proscription of the celebration of the full liturgy during the weekdays of Great Lent. This was the tradition of the early church. There are even somewhat later canons that forbid it, though the Roman Catholic acceptance of the Pendecti (or Quinisext) Council has not been consistent over the years. The same St. Gregory, Pope of Rome, referred to above is credited with recognizing a need among the faithful to commune regularly during Great Lent and therefore writing the Liturgy of the Presanctified Gifts. This liturgy is still used by the Orthodox during Great Lent, while among Roman Catholics the proscription of the regular mass is no longer observed.
I would also like to add that good source material for understanding common differences would be any primary source surrounding the Council of Florence, during which Mark of Ephesus on the Orthodox side was able to show that the very manuscripts of many church fathers found in the west did not match any of those found in the east on several important doctrinal issues. This is important, as it sheds light on why at least some doctrinal differences may have started.
Finally, it is common to list 1054 as the date of the rift between the Roman Patriarchate and the other four in the east. While this is handy, it isn't accurate. The Roman patriarchate was in and out of communion with the east just prior to this date and for about a century after. Differences in practice extended much further back, as well.
The problem with all these analyses is that they erroneously conflate the fundamentals of the orthodox faith with theological systems, theological opinions, ecclesiology and rules of the Church. The fundamentals of the orthodox faith as proclaimed by the first seven ecumenical councils are fundamentally the same as the fundamentals of the Latin Church (Roman Catholic Church). The two churches have different viewpoints with respect to theological systems, theological opinions, ecclesiology, and rules of the church.
With respect to the filioque controversy, one prominent orthodox theologian (Metropolitan Kallistos aka Timothy Ware) believes that this controversy is over linguistics and not over doctrine. In latin, there is a distinction between "filioque" and "et ex filio". "Filioque" means through the Son as from One Principle. The Roman Catholic Church believes that the Holy Spirit proceeds from the Father through the Son as from One Principle.
The real difference involves the role of the Pope. We orthodox believe that the Pope is the first among equals and that he has no authority to over-rule an ecumenical council.
Robert Haas really nailed it. Much of the reported theological differences concern language and not reality. Catholics do believe that the Pope may speak infallibility under certain and very specific conditions. Although even here when we start to get into the theological specifics the conflict with Orthodox theology is arguably still only a question of semantics.
Anyway since Haas' answer was so good the only reason I am really posting is to point out that the Catholic Church is actually comprised of 23 separate Church traditions. The Latin Church is merely one of these traditions. Among the others are the Greek Catholics, Byzantine rites of various national Catholic Churches and the Melkite Greek Catholic Church (which is the same in all respects to the Antiochian Orthodox Church except in matters concerning Church authority and as a consequence of that is in communion with Rome).
In other words what I am saying is that the theological and cultural distinctions between the Roman Catholic Church and the Greek Orthodox Church are already contained within the Catholic Church as a whole through her Eastern Rite Churches, and often when people try to contrast Orthodox theology with Catholic theology they are actually making a false dichotomy.
In the end I am tempted to say that only real difference is that Orthodox Churches are not in communion with Rome, but when we look at the situation historically even this is seldom true. In fact finding a moment of actual schism between the two traditions has been impossible to locate definitely.
Having said that let me revise my answer. I am not sure what you mean by the Greek Orthodox Church. For example did you actually mean the actual National Orthodox Church of Greece or were you simply speaking of an Orthodox Church in the Greek tradition such as the Antiochian Orthodox Church? In an case you can contrast the difference between those Churches culturally, theologically and liturgically with the same differences you will find between various Catholic Churches, for example the Roman Catholic Church with the Greek Byzantine Catholic Church or the Roman Catholic Church and the Melkite Greek Catholic Church.
What remains then is this belief that we are not in communion with each other, although when and where this actually took place specifically neither of us can say for sure. So even this confusion we share. We share everything except for the communion meal.
As for the more pragmatic differences in case your question was more pedestrian, they comprise things such as the use of Holy Water in the Roman Church and kissing of icons in the Byzantine Church. Romans cross themselves right to left with their middle finger while Byzantines marry the thumb to the index and middle finger and cross themselves left to right. Byzantines call the Mass the Divine Liturgy and use a variety of forms through out the year such as the liturgy of St. John Chrysostom and the Liturgy of St. Basil, these involve different liturgical action than does the Latin Mass Rite and are usually longer. We have different Feast days and may celebrate Easter on different days. Byzantines Baptize and Christmate their children at the same time, ideally 40 days after birth and offer them the Eucharist from that point on. Catholics sometimes Baptize their children eight days after birth (and sometimes sadly, at the other extreme, not at all), but defer the Eucharist, which we call Communion, until after the age of reason. I believe this currently 7 years old. Chrismation comes later and we call it Confirmation, except for adult converts who may receive Baptism, Confirmation and Communion all on the same day, usually at the Easter Vigil Mass.
The list in fact is quite long but this gives you a sample of the types of distinctions. That is they are myriad distinctions stemming from different codes of canon law, and different cultural norms that really for a more precise answer must be addressed by more specific questions. And again, these differences hold between Catholic Churches of different Rites as much as they do for Orthodox and Catholic Churches of different Rites.
"less precise" is not a good term to use. For example: Orthodox say that the Holy Spirit stem from the "Father through the Son", whereas Catholics say that the Holy Spirit stem from the "Father and the Son", but with the same meaning like Orthodox. For your info, Pope John Paul II accepted the lack of accuracy in this expresion, which was the main issue why Catholics split from Orthodox in 1054. The Pope said this expression exactly like Orthodox in 1995 together with the Orthodox patriarch. Because of this correction, Catholics and Orthodox should be already the same church. Sir/Madam, please don't disinform people for what is precise in saying in regard to God. God wants all Christians to be united, and all Christians should contribute for this goal.