What were the doctrinal differences that caused the split of the Roman Catholic and Orthodox Catholic Church in 1054?
There are several doctrinal differences between the Western and Eastern Churches, but the filioque controversy is often cited as the cause of the Great Schism, resulting in the Eastern and Western bishops excommunicating each other.
The filioque clause was a major controversy in the church in relation to the Holy Spirit. The question is, “from whom did the Holy Spirit proceed, the Father, or the Father and the Son?” The word filioque means “and son” in Latin. It is referred to as the “filioque clause” because the phrase “and son” was added to the Nicene Creed, indicating that the Holy Spirit proceeded from the Father “and Son.” There was so much contention over this issue that it eventually led to the split between the Roman Catholic and Eastern Orthodox churches.
Filioque, Latin for "and (from) the Son", was added in Western Christianity to the Latin text of the Nicene-Constantinopolitan Creed, which also varies from the original Greek text in having the additional phrase Deum de Deo (God from God) and in using the singular "I believe" (Latin, Credo, Greek Πιστεύω) instead of the original "We believe" (Greek Πιστεύομεν), which Oriental Orthodoxy preserves.
At the 879–880 Council of Constantinople the Eastern Orthodox Church anathematized the Filioque phrase, "as a novelty and augmentation of the Creed", and in their 1848 encyclical the Eastern Patriarchs spoke of it as a heresy. Eastern theologians argue that the Filioque clause is symptomatic of a fatal flaw in the Western understanding, which they attribute to the influence of Augustine and, by extension, to that of Thomas Aquinas. https://en.wikipedia.org/wiki/East%E2%80%93West_Schism#Differences_underlying_the_schism
Another theological issue is the Western Catholic doctrine of the Immaculate Conception which declares that the Virgin Mary was protected by God from original sin. The Orthodox Church simply acknowledges that Mary was chosen to bear Christ because she had found favour with God because of her purity and obedience. The Eastern Orthodox Church does not appear to agree with Mary being born without sin.
There may be some differences with regard to the doctrine of Damnation although they appear to be very subtle.
To all intents and purposes, the critical issue was the question, “from whom did the Holy Spirit proceed, the Father, or the Father and the Son?”
The summary in Wikipedia is a great starting point. Here are the main reasons.
A series of ecclesiastical differences and theological disputes between the Greek East and Latin West preceded the formal split that occurred in 1054.1 Prominent among these were the procession of the Holy Spirit (Filioque), whether leavened or unleavened bread should be used in the Eucharist,[a] the bishop of Rome's claim to universal jurisdiction, and the place of the See of Constantinople in relation to the pentarchy. -Wiki-
What was first in the series of causes?
In 1053, the first action was taken that would lead to a formal schism: the Greek churches in southern Italy were required to conform to Latin practices, under threat of closure. -ibid-
This first DOCTRINAL issue was the question about whether to use unleavened bread (Latin practice, Catholic) or leavened bread (Greek, Orthodox). The Orthodox follow the Gospel of John that Passover was yet to come AFTER Christ's crucifixion, hence use leavened bread. The Catholic view is to follow the Synoptics that Passover came BEFORE Christ's crucifixion, hence use unleavened bread. This controversy dates back to the very beginnings of the church.
Here's the pertinent part of the Bull of Excommunication sent by the West to the East in response to the East's failure to repent (over doctrinal issues).
For these errors and many others committed by them, Michael himself, although admonished by the letters of our lord Pope Leo, contemptuously refused to repent. Furthermore, when we, the Pope’s ambassadors, wanted to eliminate the causes of such great evils in a reasonable way, he denied us his presence and conversation, forbid churches to celebrate Mass, just as he had earlier closed the churches of the Latins and, calling them “Azymites,” had persecuted the Latins everywhere in word and deed. Bull of Excommunication
In case the point is missed, the East was calling the West dead. Azymites, meaning without leaven. So, why follow the doctrinally dead, regardless of its claims for itself?
It reached the climax when each side declared each others' priests and offerings null and void based on each's interpreations of the four gospels.
In popular opinion, the flour and water wafers of the "Franks" were not bread; their sacrifices were invalid; they were Jews not Christians. Their lifeless bread could only symbolize a soulless Christ; therefore, they had clearly fallen into the heresy of Apollinaris. The controversy became a key factor in producing the East–West Schism, which persists to this day. -source-
To continue the narrative from Wiki,
In retaliation, Patriarch Michael I Cerularius of Constantinople ordered the closure of all Latin churches in Constantinople. In 1054, the papal legate sent by Leo IX travelled to Constantinople in order, among other things, to deny Cerularius the title of "ecumenical patriarch" and insist that he recognize the pope's claim to be the head of all of the churches
In other words, Cerularius declared the Latin churches invalid, dead, doctrinally heretical. Pope Lex IX denied Cerularius' orders and insisted that the East recognize the West's claim to be head of all the churches.
From there, the Filioque question and others came into play. Cut off the head and question all of the rest.
So, to answer the OP, the initial doctrinal difference of whether to use leavened or unleavened bread may seem to modern people as much ado about nothing, although as mentioned it had simmered for some 1,000 years, the doctrinal cause was their interpretations of the Bible and then the question of who has authority about such. It was a short step for each to declare the other null and void. And then raise additional doctrinal questions.