The legal system of the Byzantine empire was continuous with that of the earlier Roman empire. The Justinian Code was a compilation of the opinion of earlier Latin jurists, beginning with the earliest known records of Roman law. Opinions of famous Roman jurists such as Ulpian and Paul, and rescripts issued by the office of the ab espistulis from the beginning of the Roman  Empire, as well as opinions mentioning earlier praetorian edicts were incorporated in the later legal codes. Law, as a system based on precedent, is therefore a major source of continuity.

What is one example of discontinuity between the Byzantine empire and the original Roman empire?

Many of the discontinuities evolved gradually as the east separated from the west, rather than being sharp ruptures. Some of the major changes were:

The Greek east, ruled from Constantinople, spoke Greek and the Latin west, ruled from Rome,  spoke Latin.

Although the Eastern Schism did not occur until much later, the eastern church and the Roman church had differences, leading to e.g. the earlier Photian schism, over such issues as tonsures, leavened vs. unleavened bread in the Eucharist, the filioque (6th century on), Illyricum, icons, and whether the Bishop of Rome was a member of the Pentarchy or had the right to dictate to all of Christendom under the title of “Pope”.