Okay, just to preface this. I'm still working towards my degree, and this is like my third comment on this sub, so more established historians please critique and correct me! I'm also not too experienced in Islamic nations, so someone else should be able to handle the subject better than myself. Anyways, on to my answer.

As we well know, Rome was a militaristic nation, and if I were to list all the civil wars I would probably exceed the Reddit Character limit(assuming it has one). But honestly, if you want a list of the civil wars in Rome, you could just look at Wikipedia, so it would be pointless me doing so. But I'll explain the key events to reflect that, in my opinion, Rome and thus the Byzantines, were prone to civil unrest.

To discuss this topic, I feel that first, we must examine the history of Rome, I'll try to be brief as we have a lot of ground to cover. As we know, the Roman Republic was founded on usurpation, with the Republic being born from a desire to remove the monarchy. Titus Livius is a famous Roman historian who covered many of the foundation myths in his book Ab Urbe Condita. Now, obviously the validity of ancient sources is dubious at best, But historian Tim Cornell summarises it excellently with his book The Beginnings of Rome. Although, monarchy in Rome is a heavily debated subjected, as it's very much shrouded in myth. The main piece of evidence we have is the inscriptions upon the Lapis Niger. Other good reading on the foundation of Rome is A History of the Roman People by Allen Ward and others. We know the republic was founded in 509 BCE, and founded upon usurpation it was.

Then we have the Roman Empire being founded upon Augustus consolidating his power after defeating Antony and Cleopatra. Obviously the events were more complex, but upon putting down his opposition, Augustus managed to become Augustus Caeser, and his rule over Rome was uncontested. He ushered in an era of relative peace known as the Pax Augusta. Though after his death this ended, and civil wars were rife once again.

Fast-Forward to the Civil Wars of the Tetrarchy. This was a series of conflicts between the co-emperors of Rome. The reason for Rome having multiple emperors is a complicated topic that would take me too long to get into now aha. But essentially after the Crisis of The Third Century, Rome had two Emperors. This is another complicated period of History, so I can't go too in-depth right now. But essentially, these wars concluded upon Constantine I's victory over Licinius at the Battle of Chrysopolis in 324CE. Good sources on these topics include History of the Later Roman Empire by Stephen Mitchell and Diocletian and the Tetrarchy by Roger Rees. 6 years after this battle, Constantine moved the capital of Rome to Byzantium, and renamed it Constantinople(see Paul Stephenson, Constantine: Unconquered Emperor. Page 190). As we can now see, The Byzantine Empire was also founded upon conflict, and upon further perusal, their list of civil conflicts is just as great as it's predecessor.

But, we know many nations were founded on war and usurpation. England, for example, was united through war. AEthelstan was coronated King of the English and then proceeded to conquer the north, reclaiming it from Danish rule(See Sarah Foot's work Æthelstan: The First King of England). This would have been a century before the Norman invasion of 1066. Now, we have a nation also founded upon war and eventually being usurped. But what makes it different from Rome? As we know, England had a few civil wars, the Barons War, War of the Roses, and The English Civil War. There were a couple more, but not many. And compared to Rome and the Byzantine's list, it is nothing.

So, the question is, what makes Rome and it's descendants so prone to civil unrest? I believe it to be a combination of factors, and I believe it to be the simple fact that, as a nation, Rome was extremely militaristic. This was a society where the strong prevailed. Individuals within the Roman Military could acquire power too easily. Julius Caeser, for example, had been campaigning with his armies for many years, which in turn led to them placing their loyalty to Caeser, and not the Republic. You have to remember these men had been campaigning with Caaeser throughout Gaul for years, to them, he was already the number one man in charge, and thus following him across the Rubicon was the obvious decision. In Rome, it was too easy for singular individuals to acquire the support of thousands. A big factor for this, at least in my opinion, is the fact that the Roman Line of succession wasn't hereditary, as hereditary succession would be too much like a monarchy(though, in my opinion, the empire was just a democratic monarchy anyways). Thus, by inheriting this system, and even codifying it into law( see The Code of Justinian) The Byzantine Empire inherited their weaknesses, leading to civil unrest being unavoidable. You also have to think about how Julius Caeser was deified, being posthumously granted the title Divus Julius, or The Divine Julius. This would have made the concept of usurpation a tempting one, as many future emperors obviously looked up to Caeser, as it became a name granted to designated Emperors to be.

Compare this to England for example, a Hereditary monarchy meant it was clear who would inherit the throne, in combination with religious views, such as The Divine Right of Kings, which means that The King answered only to God, and no earthly being held authority over The King. Thus, Civil Unrest was unlikely. Rome also did deify its emperors, however, the lack of succession clarity and their absolute refusal to be seen as a monarchy are reasons why Rome and thus The Byzantines, would not have had the same view in regards to the infallibility of their rulers.

To summarise, Rome and thus the Byzantines had a history of civil unrest, they were empires where it was too easy for an individual to consolidate power, and thus civil unrest was unavoidable. In nations such as England however, individuals had a more difficult time consolidating this power, civil unrest was much less likely. It all really boiled down to the system of government that was in place. Even the absolute mess that was the Holy Roman Empire was more stable.

I hope I answered your question well, as I said, I know nothing of Islamic History, so someone else may have to comment on that. But I hope I gave a good enough explanation of why Rome and The Byzantines were seemingly in a constant state of unrest compared to other nations. I also have to go sort something out so I'll edit in a full bibliography later, but most my sources are already within my answer.

Bibliography:

Primary Sources:

Titus Livius- Ab Urbe Condita

The Lapis Niger

The Code of Justinian

Secondary Sources:

Stephen Mitchell- History of the Later Roman Empire

Roger Rees- Diocletian and the Tetrarchy

Sarah Foot- Æthelstan: The First King of England

Paul Stephenson- Constantine: Unconquered Emperor

National Geographic Article on Caeser