There is great continuity, because under Constantine, the Empire moved East. So, there is great continuity in terms of the stock of people, worldview, and ethos. In addition, one of the greatest continuities comes from the use of Roman law. Roman law is one of the most important developments and contributions of the Romans to humanity. In fact, apart from the Bible, the second most important book in the world might be Justinian's Digest.
Just a point of clarification: Diocletian is the emperor who actually first divided the empire into Western and Eastern halves. Constantine reunited it, and then divided it again. Theodosius did the same thing, and his division, which took place according to his will, was the final division between east and west.
The Roman Empire was divided in two parts by Constantine; some historians assert this was a bad decision on Constantine's part. Rome remained the center of the western and Latin speaking part of the Empire while Byzantium/Constantinople became the center of power of the eastern and Greek speaking part. The original continuity was that after the fall of the city of Rome (476 AD), the Roman empire and culture continued in Byzantium. The greatest discontinuity is that a divergence grew up between how the West practiced Christianity and how the East practiced it. This difference radiates throughout time even to today.
During the Roman Republic and well into the Roman Empire, trade became firmly established around the Mediterranean. However, unlike in the West, the East continued to trade after the Western Fall -- that fact alone implies that the political and social and economic organization in the East remained intact. Because the East continued trade, they continued to exist, although weakening over time until the 1400's when Byzantium finally fell.
One other way in which there was continuity was in the minds of the people. If you would have asked a person in the Byzantine Empire if they were Roman, they would have said they were. To them, there was continuity. The Byzantines also continued to use Roman law that they had codified in the time of Justinian.
It was a continuity in that it maintained the basic state structure of the late Roman Empire, including an emperor, a Senate (albeit one with mostly ceremonial duties,) and a complex bureaucracy staffed by officials dependent on the emperor for patronage. During the early years of the Byzantine Empire the emperors, most famously Justinian, also pursued an aggressive,militaristic policy in expanding throughout the Mediterranean basin. It was different, however, in that the language and culture within the Empire were predominately Greek, and the state religion was Orthodox Christian.
Why were there continuities or discontinuities?