Post-Roman Europe and the Byzantine Empire are often defined by difference, and they certainly considered one another foreign cultures. But similarities did exist and are worth exploring.
Before beginning, however, there is one crucial difference to address. For most of its history, the Byzantine Empire was a unified state, while western Europe in the same period was not. Europe consisted of multiple nations, often at violent odds with one another. There is no single "western Europe" to which the Byzantine Empire can be compared as a political entity.
That said, the European nations did have several qualities in common with one another and with the Byzantine Empire.
Politically, both cultures were theocratic and authoritarian. As already stated, the Byzantine Empire had one political system, while western Europe had dozens, but virtually all had the quality in common of being ruled by powerful monarchs vested with political, military, and religious authority. Crucially, both the European monarchies and the Byzantine Empire were Christian and viewed their rule as being the will of God. A deep and longstanding divide existed between European Roman Catholicism and Byzantine Orthodox Christianity, but each group did consider one another Christian and historically had a sense of religious allegiance, not to mention a shared enmity toward neighboring powers espousing Islam, Judaism, or other faiths. An example can be seen in the First Crusade, which began when the Byzantine Emperor Alexios I called for, and received, military support from western Europe for military support against the Muslim Seljuqs on the basis of their shared Christianity.
Socially, one major similarity between European and Byzantine culture was that they considered themselves descendants of the Roman Empire. Again, in western Europe dozens of different interpretations of that descent existed, but "being Roman" was a consistent goal: everyone from the Carolingian empire of 800 CE to Fascist Italy in 1922 presented themselves as, and in most cases genuinely believed themselves to be, heirs to ancient Rome. So too did the Byzantines, who represented a continuation of the original Eastern Roman Empire, which had been an established body since the Roman Emperor Constantine I moved the seat of the Empire to the Byzantine capital of Constantinople (named in his honor after his death) in 330 CE. The western Europeans and Byzantines had different visions of what Rome was and should become, but both modeled their society on Rome as they understood it.
Economic similarities between Europe and the Byzantine Empire also existed. The economic relationship between the two was extremely complex, and their relative fortunes waxed and waned over hundreds of years of history. What they consistently had in common was an emphasis on trade and activity. Both European and Byzantine traders ranged over the medieval world seeking new opportunities, and both attached value to novelty that was less prevalent in other cultures.
Culturally, the vital shared quality between Europe and Byzantium was their Christianity. As already noted, while significant hostility often existed between the two branches of Christianity, as shown in the history of the Fourth Crusade, Roman Catholicism and Orthodox Christianity generally acknowledged a shared origin and tradition. Both also valued Greek and Roman culture, especially after conquests in Muslim Spain in 1100–1300 CE reintroduced much Greek thought to western Europe.
Similarities certainly existed between western Europe and the Byzantine Empire in the years of their coexistence. At the same time, profound differences also separated the two. Taking both into account is vital to understanding European history at the time.