Conquest:
There were several very important phases of Byzantine conquest. 1) 533-554: Justinian's reign was successful in taking Italy, Sicily, North Africa. 2) Basil II and the Macedonian Dynasty (920-1042) were successful in retaking and conquering the Bulgar Kingdom, Armenia, parts of Levant, and the Danube is restored as the Empire's northern border. 3) The Komnenos Dynasty (1081-1180) restores Byzantine control to the shores of Asia Minor and interior Asia Minor. Now, it goes without saying that there were periods of decline and reversal. However, each major period of decline was followed by a restoration (for the most part). Total restoration? No, but partially each time. On the other hand the Athenians lost one war and that was it.. dust pan of history. Even after the fall of Constantinople in 1204 and loss of the city for sixty years it was retaken. This however did lead to the final period of decline and eventual fall to the Ottoman Turks.
https://en.wikipedia.org/wiki/Timeline_of_the_Byzantine_Empire
Religion:
The Great Schism was a religious schism between Rome and Constantinople. This event led to the separation of the Christian faith into two camps: Eastern Orthodox and Catholicism. There are up to 300 million Eastern Orthodox worshipers today. There are 1.2 billion practicing Catholics worldwide. http://www.bbc.com/news/world-21443313 and https://oca.org/questions/namerica/how-many-orthodox
How many practicing ancient Greek Pagans are they worldwide? Near zero I would imagine. Through this religious event between the Byzantine Empire and the Vatican we can see the formation of two of the world's major religious systems. The religious effects of the Byzantine Empire and this specific event continue to shape our modern world.
https://en.wikipedia.org/wiki/East%E2%80%93West_Schism
Law:
The Byzantine Empire preserved ancient Roman law, but also made their own codex, improvements, and new laws. For example in 438 Theodosius published a codex of 16 books of all standing laws from the Age of Constantine I to the present. But, in 518 Justinian began a decade long practice of codifying, expanding, improving, studying, and commissioning a series of commissions to develop a new legal code for the Empire. This led to the Corpus Iuris Civills which (basically) remained as the Empire's legal system until the fall of Constantinople. Now, there were other additions and improvements to these laws such as the Basillcs of Leo and the Farmer's Law (and so forth). The most important fact is that these legal codes and laws put forward by the Byzantine Empire INFLUENCED EVERYTHING. Legal traditions in Eastern Europe are still influenced by the Farmer's Law. Sharia Law was influenced by the Byzantine Empire, Anglo-American common law was influenced by the Byzantine Empire. Many of the world's current legal systems were influenced by the code but forward by Justinian. They still are TODAY.
https://en.wikipedia.org/wiki/Byzantine_law
Longevity:
A pretty basic one. No one can dispute that the Athenian Empire outlasted the Byzantine Empire. The Athenian Empire lasted a little more than a hundred years. They lost the Peloponnesian War. After that their Empire was done. The Byzantine Empire went through long periods of military success and conquest and then declined only to reverse their fortunes on numerous occasions. They lasted over a thousand years. The empire held for over a thousand years. No comparison.
Other Achievements:
A major Byzantine invention was Greek fire. This weapon was unique for the ancient world. This weapon would be used during naval battles. Even today we don’t know exactly how they used this weapon during combat and what its composition was. However, at the time and when the Byzantine’s deployed it, well it was a game changer for naval combat. It is also said that the Byzantine’s developed flamethrowers and grenades. Furthermore they engineered siege equipment and innovated siege warfare with the hand trebuchet and counterweight trebuchet. Let’s not forget that they were master engineers as well that built the Hagia Sophia, Pendentive dome, Pointed arch bridge, ship mill, cross-in-square, and little Hagia Sophia.
http://www.ranker.com/list/list-of-byzantine-inventions/wayne-szalinski
Constantinople:
For our purposes we will be looking at Constantinople after 395. This includes the construction of the legendary Theodosian Walls built in 404/405. These walls held all invaders until 1204. These walls and fortifications protected the city of Constantinople for 800 years. Athens, for example, was burnt to the ground during the Greco-Persian Wars. We’ve already discussed the Hagia Sophia, which was a historical marvel, but there were other institutions in Constantinople worthy of mention. The Golden Gate and land walls for example or the University of Constantinople which was founded in 425 AD. (I am omitting institutions founded before 395), but I think it is worthy to note institutions such as the Imperial Library which contained the relics of the Library of Alexandria and preserved this knowledge until 1204 and 1453. This next graph will compare population size for a capital city. The population of a capital is indicative of the strength of the Empire as a whole. There are several different variations of this, but you can see for yourself. In 500, there is little dispute that Constantinople was the largest city in the world. Constantinople shows up on this list several times thereafter. It shows up tied for largest in 600 AD. It shows up again as a major city in 1000-1050. It shows up again from 1127-1153. Athens isn’t even on this city-population metric even at its height. As a city-state it is a backwater (at least concerning population) to the other great cities of the 5th Century BC to that there is no dispute.
https://en.wikipedia.org/wiki/List_of_largest_cities_throughout_history
Conclusion;
Through religion and law the Byzantine Empire still shapes the MODERN world. Through conquest and longevity the Byzantine Empire prolonged the ancient Roman Empire into the late Medieval period. The great city of Constantinople for portions of its history was unmatched and the largest center of civilization in the entire world. Through inventions and achievements new technologies and weapons were introduced (some of which can’t be figured out by our scientists today). Folks, this is the A-Team Empire. Athens is a D- Empire at best. Unfortunately though the US public education system is skewed towards Greece and Rome. I understand why people who are not familiar with this empire or its history think Athens (a small, remote, backwater) is a better Empire. It’s strange and bewildering, but I blame modern education. I know that competent historians would agree with me on this and that is all that matters to me.