Centuries of political and religious conflict, basically. Without getting into the nitty-gritty details, many of which have entire books about them, I'll focus on the things that caused them to be distanced.

1. Who governs the church? Catholics, as I'm sure you know, believe in a doctrine called Papal Supremacy. This doctrine, based on the apostolic succession from Peter to whatever Pope was ruling at the time and the fact that Peter was "the rock upon which I will build my Church," meant that the popes viewed themselves as not just the head of the western Christians, but of all Christendom. This directly flew in the face of the the eastern view of church governance, namely- Caesaropapism and the Pentarchy. The Pentarchy, established by Justinian, was a system where the five great patriarchs (Alexandria, Antioch, Constantinople, Jerusalem, and Rome) would be the religious governors, in a way, of their respective provinces (a nice map can be found here.) The five patriarchs were expected to pretty much leave eachother alone and the five could convene councils to deal with doctrinal issues. The pope was indeed recognized as the leader of these five, but as a "first among equals," not as a supreme power. Caesaropapism was the idea (one that was coined some hundred years after as a way to describe the Byzantine church) that the Emperor, as God's viceroy on Earth, was the Supreme authority on earth, including over religious affairs. This is reflected in the fact that the Emperors would often overthrow the Patriarchs of Consantinople whenever there was an issue, and the Church (in the East mostly, although there is a period of time called the Byzantine Papacy where the Emperors directly chose the Popes) was generally at the whim of the Emperor, as seen in episodes such as the proclamation of the Henotikon by Zeno, Leo III starting iconoclasm (which I'll get into in just a moment!).

2. Political separation. The Popes of Rome, especially after the overthrow of the Exarchate of Ravenna, which was the primary way that the Byzantines held power over the Pope, were simply in a different world than the Byzantines. They were surrounded by Franks and other "Barbarians," so the whims of the Emperor simply weren't all that relevant to the governance of the Western Church. The West was developing with their own Empire, the Frankish and later Holy Roman one, while the East was still "Roman" in its subordination to the true Roman Emperor. The Franks were quite happy to support Papal Supremacy, as it meant that "their" patriarch was the power over the Byzantines.

3. Theological disputes. The Church in the West was, for the most part, not particularly subjected to wide-scale theological disputes such as the east was. In the west, arianism was really only practiced by the Barbarians themselves, not their "Roman" subjects who generally followed the Catholic faith. As such, the West became very "set in its ways" so to say, and generally the Pope's were champions of mainstream traditional christian thought. The east, on the other hand, consistently dealt with wide scale heresies. The Monophysites, in particular, were responded to in a variety of ways such as the Henotikon and the institution of Monothelitism, which were seen as heretical by the West. This caused great strain in the church, something that was exacerbated by the iconoclast controversy. The West was once more a supporter of orthodoxy, preserving icons, while the east went through trial and tribulation until the eventual removal of iconoclasm, but the damage was done and the west and easy very much were already acting independently of one another.

The biggest theological issue was the insertion of the Filoque into the Nicene Creed. Without getting too much into the theological aspects of it, it basically added the phrase "and from the son," in the description of the trinity in the nicene creed, making:

And in the Holy Spirit, the Lord, the giver of life,

who proceeds from the Father ⟨and the Son⟩.

who with the Father and the Son is adored and glorified.

The East was not happy about this, as in their view this was underestimating the power of the Father in the Trinity.

Also the West used unleavened bread, which the greeks were not fans of. You would need to consult with a Theologian for that, though.

So all of these things are sort of cooking in a stew of resentment, needing something to set the powder keg off. This came when churches in southern Italy and Sicily, once under Byzantine theological rule but now, were conquered by the Normans who followed the Pope and placed the Byzantine churches under his rule. The churches were forced closed unless they agreed to follow western practices (Priestly celibacy, filoque, unleavened bread, etc) and in response Michael I Cerularius closed all of the latin-rite churches in Constantinople. Furthermore, much the chagrin of the West, he adopted the title of "Ecumenical Patriarch." Ecumenical, meaning universal, seemed to say that he was the head of all Christians (he knew he had no way to enforce this) while the Pope, furious all the way, claimed that for himself and refused to allow him have that title. A flurry of angry letters were exchanged until eventually Cardinal Humbert, the Papal legate, excommunicated Michael and Michael excommunicated him right back, in a way formalizing the division of the Church.

Further events would exacerbate this division, such as the Western sack of Constantinople in 1204, along with multiple attempts to restore the unity by the Byzantines (they were in desparate need of Western support and figured the Pope could help them get it), but this just made the Greeks angry. All hopes of unity were finally destroyed when the Ottomans captured Constantinople. They were in no way interested in seeing their Christian subjects become the subjects of a foreign religious power, and so Mehmed the Conquerer appointed Gennadius, basically the most anti-western guy he could find, as Patriarch and put it under Ottoman jurisdiction. Thus, you have the system of two church's with little theological difference you have today!

On the subject I very much like the book "Christianity: The East/West Divide: The Great Schism." It's a popular history more than a scholarly source, but it does the job well.