The primary difference was over the authority of the Pope and the use of Icons. The Byzantine church at one time considered icons to be graven images, and worked to abolish them completely. This was the famous "iconoclastic controversy." This controversy never erupted in the Roman church.
Another issue was the authority of the Pope. The Popes claimed to be the vicar of Christ on earth, and drew their authority from Jesus' words to St. Peter, whom they considered to be the first Pope:
And I tell you that you are Peter, and on this rock I will build my church, and the gates of Hades will not overcome it. I will give you the keys of the kingdom of heaven; whatever you bind on earth will be bound in heaven, and whatever you loose on earth will be loosed in heaven.”
A long and difficult struggle between Popes and rulers in Europe actually prevented the establishment of a unified empire in Europe for many years. This was not the case in Byzantium where the Emperor himself was both head of the church and head of state, under a doctrine known as Caesaropapism. As a result, the disputes common in the Roman Church never appeared in the Byzantine Church.
The Byzantine Church tended to be more traditional in its beliefs, and hence became known as the "orthodox" church, the term literally meaning "traditional." The Roman Church spread much further and thus became known as the Catholic or "universal" church. Hence there is today the Roman Catholic Church and the Greek Orthodox Church.