The Byzantines reacted more efficiently to the crisis at hand. For the Sassanids, I cannot say, but the civil war suggested by The_Magic is definitively one of the main reasons.

As for the Byzantines:

One of the most important contributing factors was the establishment of the Theme-system in Anatolia. As the Byzantine armies retreated from south and east of the empire in the face of the Arab onslaught, the Themes were formed by combining smaller, older provinces around where each of these armies was stationed in. Themes combined military and civilian political authority in the hands of the military governor, the Strategos. Each Theme could field a substantial force of soldiers, thus making it easy for the governors to respond to Arab raids.

This new defensive arrangement worked reasonably well for the Byzantines as Arab raids and military campaigns could be more effectively countered in the mountainous terrain of Anatolia and in the narrow passes of Taurus mountains, which led in and out of central Anatolia. However, it also also had some negative effects for the Byzantines as it affected the quality of their foor soldiers, turning them into a defensive militia whose task was to garrison cities and man mountain passes. The striking power of the Byzantine army was, at this point, in its cavalry force, whose task was to screen and intercept Arab raiders. Constant Arab raids reduced the cities of Asia minor into fortified military garrisons to which the rural population would flee when a raid occurred. This made the rural areas more defendable, yet it all but erased traces of city culture and commerce in the Anatolia.

The Arabs tried to siege Constantinople two times: in 674-678 and in 717-718. The success of either of these sieges would have certainly all but extinguished the empire in one single blow as Constantinople and Thessalonike were, at that point, the only "real" cities in the empire. These sieges failed due to the walls of Constantinople, Byzantine naval victories (greek fire) and Byzantine military and diplomatic successes. I could go into detail with these if you want. Nevertheless, the Imperial capital remained in Byzantine hands at all times, which brings me to the last factor that contributed to Byzantine survival.

The Imperial government never ceased to function. Although the empire was greatly diminished with the Arab invasions, it could still collect taxes very effectively, allowing it to field comparably large armies and navies against the Arab invaders. The Theme system further contributed to this. In short, we can say that the Byzantine state itself worked much more effectively than its Arab counterpart.

Thus we can conclude that what mattered in Byzantine survival was the adoption of the Theme system and the longevity and effectiveness of Byzantine state apparatus, which yielded benefits to both the empire's civilian and military administration.

This was my first time posting here, I hope this answered your question! For sources and reading, I can recommend Warfare, State and Society in the Byzantine World 565-1204 and Byzantium: A History, both by John Haldon. For contemporary sources, the english translation of Leo VI's Taktika by George T. Dennis (The Taktika of Leo VI) is recommendable, among others. I am currently in the process of preparing a Bachelor's Thesis on the Taktika of Leo VI.

The Sassanids were further weakened due to civil war. I believe there were something like 13 claimants to the Persian throne after their war with Byzantium.

A really nifty source about the Sasanians and their conquest by the Arabs is Decline and Fall of the Sasanian Empire by Parvaneh Pourshariati. She contends that the Sasanian Empire was never really a centralized state at all, but rather a confederacy between Sasanians and Parthians (ruling elites from different parts of Iran). At the time of the Arab conquests, that confederacy was beginning to fray significantly. The Sasanians were trying to centralize their grip on power, and the local Parthian nobilities were trying to resist. As a result, when the Arabs came, there wasn't really any sort of united front that could stop them.

(As it happens, the same internal strife had helped the Byzantines win the last big war between the two Classical states, a war which had in turn exhausted both of them and left them vulnerable to outside attack. The religious strife that was tearing at the Eastern Roman Empire was another big factor in why Egypt and the Near Eastern areas were quick to fall to the Arabs, but that's a whole other question.)

Anyway, an interesting book, one of the best I've read on the Sasanian Empire specifically. Worth checking out.