This was actually a pretty slow process. You have to remember that the two churches didn't even officially split until 1054 CE. Lets refer to the Christian Church in Europe before 1054 as Chalcedonian Christianity, and the Churches afterwards as Catholic and Orthodox Christianity.

So, right up until 1054, Chalcedonian Christianity allegedly was governed by laws laid down at the various ecumenical councils of the church and administered at a local level by Bishops, Metropolitans, and parish Priests. Overarching all of this were five Patriarchs, the Bishops of Rome, Antioch, Constantinople, Alexandria, and Jerusalem. This system was called the Pentarchy (Pent- five, -archy rule of) Of these, the Bishop of Rome was given a good deal of respect and deference by the four others, but by no means was his authority final. Today we call the Bishop of Rome the Pope.

Anyway, as time went on, Chalcedonian Christianity increasingly adapted itself to the local regions in which it was practiced, e.g. in Ireland Christianity was radically different than it was in Tunisia. Furthermore, in the 7th and 8th centuries, Antioch, Jerusalem, and Alexandria were seized by Islamic armies, meaning that their power and influence in the pentarchy declined. Furthermore, the rise of the Holy Roman Empire in the West relied on the Bishop of Rome for authority and prestige, just as the Eastern Roman Empire relied on the Bishop of Constantinople.

This pitted Rome against Constantinople in struggles for both secular and spiritual power. Several attempts were made to build bridges, but few succeeded. By 1054, the Pope had essentially had enough, and sent a Papal legate to Constantinople to deny the new Bishop there the title of "Ecumenical Patriarch". This was actually way more complicated than I am portraying it to be, things got very hairy.

By any means, after that point, Christianity in the West and East developed very differently. In the West, throughout the 10th, 11th, and 12th centuries Popes increasingly centralized control of Europe under the (now) Catholic Church. They developed a theory of power called Plenitudo Potestatis, which essentially held that as the Vicar of Christ the Pope held both spiritual power (sacerdotium), and secular power (regnum), over the entire world, even over heretics, infidels, and people who had never heard the gospel. This philosophy of power remained in force until the death of Innocent IV in 1252, after whose reign fewer and fewer secular rulers deferred to the papacy.

In the East, Orthodox Christianity developed a decentralized model, granting autocephaly (essentially self-determination or autonomy) to Bishops of various regions including Georgia, Russia, Bulgaria, and Syria. The Patriarchs in Constantinople retained control over Greece. They also extended the olive branch to previously persecuted churches including Coptics, Jacobites, and Nestorians. Essentially they created a loosely governed confederation of independent churches who claimed authority over certain aspects of spirituality in certain regions.

Now, this schism actually didn't sit well with either side, and there were manifold attempts at reconciliation. In 1245, certain Russian princes agreed to convert to Catholicism in exchange for help against the Mongols. No help actually came, so they just converted back to Orthodoxy in 1252. Both the Second Council of Lyon (1274) and the Council of Florence (1439) attempted to reintegrate Orthodoxy, to no avail.

Today, Catholicism is trying to reconcile both with Orthodoxy, and with certain Protestant sects such as the Anglican Church. As such, they grant a great deal of autonomy to sects like Byzantine Catholics (whose adherents included Andy Warhol, weirdly enough), Ethiopian Orthodox, and Indian Catholics/Nestorians. Orthodoxy, on the other hand, is very worried that its adherents will no longer differentiate between Catholicism and Orthodoxy, and certain churches are actively trying to keep people from loosely straying between the two.

As to practice, its mostly just theological minutiae. They still do most of the same sacraments, and since Vatican II they both preach in the vernacular.