The primary similarity between the two is that they both considered themselves "Roman." The people of both Empires referred to themselves as Romans, and Latin was the official language of both. Culturally, there were a number of similarities: both had a hippodrome for circuses, gladiatorial fights and horse races, and both referred to their Emperor by the title of Caesar. Neither Empire recognized a nobility, although class distinction was apparent in both.
At that point, differences abound. Although Latin was the official language of both Empires, the common language of the Byzantine Empire was Greek. Additionally, the Byzantine Emperor was also the head of the church, a policy known as Caesaropapism. In the West, the head of the Church was the Pope; and the division of power between church and state led to constant struggles between the two. No such difficulty existed in the East. Religious differences between the two led the Pope and the Archbishop of Constantinople to excommunicate each other, as a result of which in 1054 the Eastern Church separated itself from the Western Church, the Eastern Church taking the title of Orthodox (meaning traditional) and the Western Church the title of Roman Catholic (Catholic meaning universal.)
The Western Empire effectively ended with the forced abdication of Romulus II Augustulus in 476 C.E., a date considered the beginning of the European Medieval Age. That same age is considered to have ended with the fall of Constantinople (or Byzantium) to the Turks in 1453, after which it was known as Istanbul.