So obviously in terms of who would collapse first, the odds were always stacked in favour of the richer and more populous eastern half of the Roman Empire. But what if the Western Roman Empire, perhaps under a surviving Majorian for example, succeeds in not only reclaiming North Africa from the Vandals, but dealing with the traitorous Ricimer and stabalising the Western Empire. While the Eastern Empire has much poorer luck, perhaps a succession of weak rulers or regencies mixed in with incursions across the Danube and successful Sassanian invasions, both of whom divide the failing Eastern Roman empire amongst themselves.
Obviously the ramifications of this affect or too many to count and this is such a massive butterfly effect, but what would be the major cultural, political, demographic and religious consequences of such a dramatic reversal of fortunes for both halves of the Roman world?
Here's an idea: You never said the Eastern Half has to be crippled in the late 5th or early 6th centuries (that wouldn't be very easy anyway since the Empire was recovering from the disasters of the 5th century during that time).
Here's a rough timeline:
Majorian recovers the West, making for a surviving WRE. The ERE on the other hand basically follows a similar path to OTL during the late 5th and early 6th centuries. The Goths are ultimately dealt with in a joint East-West campaign and the remnants are mostly settled in the West.
Things begin to change is with Justinian. Since the West never fell, his ideological raison d'etre needs to change. While in OTL Justinian wished to restore the West, ITTL he wishes to reunify the Empire under his sole rule (simply put: One Empire, One Emperor).
During the 530s Justinian declares war on the West and easily takes North Africa in a lucky break, however, Italy is not so easily taken, just as in OTL, due to the general popularity of Majorian's dynasty. The East also has to deal with the historical Persian invasions. This is right up until the Plague strikes, basically reducing both sides' ability to fight to near nothing, making Italy a practical stalemate during the 540s. During the 550s Justinian just narrowly succeeds in taking Italy, but has little ability to do much else, beyond a partial conquest of Hispania.
The period from 565-602 is mostly the same, especially on the Eastern Front. The Western Empire, being too weakened to take Italy on their own due to the Plague, recruits the Lombards to help them. They are partially successful, but the Eastern Empire successfully controls many coastal cities and most of the South.
The 7th Century is mostly about the East being wracked by Persian invasion and the West attempting to recover their position in the West, while attempting to reign in the Lombards. After defeating the Persians and recovering the Eastern Provinces, Heraclius dies, leaving the Empire to far less competent heirs than OTL. The Arabs, instead of conquering the East as OTL, do to the East what the Germans did to the West OTL. With the East wracked by war and plague and severely depopulated, the Arabs are welcomed by the Eastern Emperors, however, the Arabs maintain their tribal cohesion and identities and take over the Levant, Egypt, and Anatolia. While the Slavs take over the Balkans in a very similar fashion. This basically recreates the West's 5th Century in the East during the 7th Century.
Basically almost the entire Eastern Empire controlled by foreign tribes and the West is both disinterested and incapable of intervening. Eventually, during the 7th century, a Slavic general does away with the the Imperial System in Constantinople.
By the end of the 7th century the Western Empire has successfully retaken Hispania, North Africa, and Italy, barely, and the early 8th century the Western Empire slowly and limply recovering economically until the final major bout of plague happens. From then on the West begins to fully recover its strength. During the 9th century sees the Reconquest of the Balkans from the Slavic Kingdom there, centered around Thessaloniki (Constantinople is largely reduced to its past as the city of Byzantium, with its city walls dismantled). They also recover Anatolia as well by the end of the 9th Century.