I've read a good deal about the ERE/Byzantines over the years, and one thing I noticed is that they almost always ended up paying tribute in exchange for peace - even when they apparently won a war.
Specifically, I'm surprised at how often they ended up paying the Persians, even though the two empires seemed to be more or less equally matched before 602. But of course, it wasn't limited to the Persians - they also payed off the Avars, Pechenegs, huns, you name it.
Again, it doesn't seem like it mattered much if they won the war, and tribute never flowed in the other direction - at least, not that I've ever noticed in my readings.
What gives?
Because they could and it was cheaper. The Byzantines, for the most part until the full adoption of the Themata system during the 7th century, relied on mercenaries or their own expensive troops, which costs a lot of money. Alongside that, the Byzantines were, for the most part, quite content with their borders (This is a theme throughout Roman History, expansion wars are pretty rare outside of Trajan in the post Julio-Claudian Roman World until Justinian) since they were mostly made up of natural, easy to guard frontiers like the Euphrates, Danube, and either deserts or seas. There simply wasn't much to gain for trying to take territory that wasn't behind their already extant defenses. Alongside that, even if you win a "war" against the Avars or the Pechenegs, they're still going to raid your territory, so paying them off was a way to help ensure that your cities and towns would be relatively safe.^1
Alongside this, a treaty where you're providing tribute is much more likely to not only make the enemy become peaceful, but to make them STAY peaceful. Part of the reason Justinian was willing to pay off the Persians, despite his relative success in his wars against them in the Iberian and Lazic wars, was because he wanted to make damn sure that they wouldn't be attacking his eastern front so he could focus on his Western Expansion.^2
Finally, they also just lost a lot of wars, particularly against the Arabs and the Bulgars. They didn't lose hard enough to get conquered, but enough that the enemy could make demands on them for peace.^3
The Byzantine Empire's economy was incredibly powerful, the Emperors could fairly reliably raise large sums of money to pay people. It was simply less destructive to just swallow the pain and pay the Huns to leave you alone than to send your army and risk annihilation. That's not to say that the Byzantines never demanded tribute of their own. In the Byzantine-Sassanian war fought by Marcian in the late 6th century, the Byzantines got to put their favorite candidate on the Persian throne and demanded much of Armenia as well. Even in the great "final" war between the Persians and the Byzantines, Heraclius demanded a fairly hefty monetary indemnity in exchange for peace (not much more though because he knew his resources were stretched thin as it was.) The Byzantines would also impose tribute on their enemies, you see this a lot in later Byzantine History under the Macedonian Emperors, who often envassalated and turned into tribute states many of the smaller states that bordered them. The Komnenoi did this as well to the states of Armenia in Cilicia and the Crusader states (In a personal favorite moment of mine, Manuel Komnenos marched into Jerusalem on a horse while the King of Jerusalem was forced to simply walk behind him.)^4