As for myself, I'm going to go back to my traditional 6 point scoring
1) Territorial conquest/political sway: Byzantines
I have a bit of a hard time giving this to the Byzantines, frankly, as they never conquered much. Most of their existence was spent defending the borders that they inherited from the Roman empire, outside of Justinian's forays into the Vandal kingdoms and Italy. Athens, meanwhile, started as just a small city-state and via its navy grew a trade/colonial empire that encompassed dozens of other city states under its sway. Make no mistake - without Athens, Greece would have fallen to the Persians.
But, as with anything, money talks. And the Byzantines had lots of money, which they used to bribe away barbarian tribes as well as buy allies in times of war. So while they didn't quite conquer anything, this political influence I will grudgingly say weighs more than that of Athens - though not by much.
2) Artistic achievements: Athens
There should be no contest with this. Athenian art and architecture was the pinnacle of art in the West until the Renaissance. Yes, the Byzantines built the Hagia Sophia and many other churches. But the Athenians built many beautiful temples and great sculptures and reliefs. This may be skewed since realistic depiction of the human form had gone out of fashion in medieval times (which is what the Byzantines are - a medieval civilization), but time also weighs in Athens' favor. Even if the Byzantines and Athens completed duplicate works, the with would go to Athens since they did it a thousand years before the Byzantines. Kinda like how artists today could recreate Michelangelo's David, but who fuckin cares?
3) Scientific achievements: Athens
Granted, I don't know all that much about Byzantine scientific achievements aside from Greek fire (which is a marvel). However, Athenian scientists laid down the foundation for so many modern fields of science: Engineering (Archimedes), Medicine (Hypocrates), Physics/Astronomy (Aristotle). Today's achievements stand on the shoulders of giants, and those giants were Athenian. Safekeeping those achievements is not on the same level as achieving them in the first place.
4) Size/longevity: Byzantines
Here ya go ND. This is the Byzantine's unquestionable win category.
5) Lasting impact on the world: Draw
So, I'm going to call this a draw. Athens represents the foundation of modern thought. Art, science, philosophy: it all starts with Athens. The Romans studied the Athenians and replicated what they could. Later on Renaissance and Enlightenment thinkers went back to these classical works. The butterfly effect is huge.
On the other hand, the only reason why these later writers had access to these works are because of the Byzantines (Well, to be correct its actually due to the Venetians plundering Constantinople at the end of the fourth crusade and bringing back lots of ancient works). By the mere fact of existing, the Byzantines kept most of Europe Christian, created the schism between Catholicism and Orthodox Christianity, as well as had a great and lasting impact on the world via Russia (who once Constantinople fell, felt that they were the next continuation of the Roman Empire).
These are not trivial things on either side, and I cannot assign a greater value to either one.
6) Notable historic figures: Athens
This is where Athens takes the lead again. I can name three people from the Byzantine Empire. Justinian, Theodora and Belisarius. I sorta know Basil and Heraclius. But besides Justinian and Belisarius, those were just some pretty decent rulers who were able to right the ever-sinking ship of the Byzantines. A good emperor, yes. But history does not deem that enough to be considered a Great Person.
The Athenians meanwhile have Pericles, Aristotle, Thucydides, Socrates, Plato, Sophocles, Eurypides, Aristophanes, Aesop, Archimedes, Euclid, Pythagoras, Hippocrates, Demosthenes, and on and on (well, actually that's most of them). These men did so much to impact their particular field of work that they are greater still known today, almost 2500 years after their death. These are the men who are Great Writers and Great Scientists in Civilization V lol.
End result: 3-2-1 in favor of Athens. ##VOTE Athens