My master's thesis has as its historical background the East/West split, and the fall of the Western Roman Empire, so I can maybe speak to other reasons for its survival. To answer your question, the Eastern benefitted from better supply networks as well as who they were in conflict with. I can speak primarily to economics and the geo- part of geopolitics, so as to provide a different perspective.

In the fourth century C.E., the capital was moved to Constantinople, which effectively halved the resources being sent to Rome and the west. For example, annona (basically, the food dole) payments from Egypt, a major producer region, were diverted to Constantinople. This diversion led to the loss of western shippers (navicularii), who aren't mentioned in any texts after the fourth century.

Meanwhile, the Levant experienced a growth surge from the fourth to the sixth century, based on their high-quality viticulture (referred to in contemporary sources as vinum optimum). Furthermore, navicularii were subsidized by the state, which allowed for goods to be easily and quickly moved from the highly productive regions of North Africa, Asia Minor, and the Levant to urban centers and military frontiers. World Systems Theory (Immanuel Wallerstein) is particularly informative when it comes to this sort of thing, though it takes some finagling in applying it to a premodern society. Furthermore, the currents in the eastern Mediterranean made it easy to ship along a north-south axis. (Mediterranean navigation and seafaring is part of my specialty, and I could talk about it all day, but I will refrain).

The fall of the Western Empire in 476 (conventional date) is roughly contemporaneous with Vandal conquest of North Africa. The Vandals stayed in the West; in fact, Mattingly and Hitchner note that by the late 5th century, the Vandal court was "a center of elite Roman-African culture with ties to Constantinople." Rather than an enemy, the Vandals appear as a partner to the Eastern Empire. This is clearly a tenuous situation, however, because by 536, Justinian takes North Africa back (resuming taxation).

While the Vandals were pressuring the western and southern Mediterranean, to the north were Huns, Avars, and Sclaveni, while to the east were Sassanian Persians. These frontiers are better described by harassment rather than all-out war, and were more easily maintained. The 6th-century Balkan frontier was held (barely) by payments to the Avars, for example. Furthermore, the multiplicity of tribes was a problem for any forces in the Danubian region. Two Gothic tribes asked for asylum from Hunnish pressure on that front, and though there was a build-up in Hunnish forces until Attila's death in 453, all-out war never came. After Atilla's death, infighting made for an easier time for Roman forces.

The Persians left Roman Syria devastated in the 3rd century C.E., but by the 5th century, a tenuous peace had been established (which erodes within 100 years).

So, while it's difficult to attribute any one thing to why the two empires' fates were so different, supply and the nature of the different conflicts are two factors.

Further reading: Curta, F. 2001. The Making of the Slavs: History and Archaeology of the Lower Danube Region, c. 500-700. Cambridge: Cambridge University Press.

Heather, P. 1995. “The Huns and the End of the Roman Empire in Western Europe.” English Historical Review 110:4-41.

Mattingly, D.J., and R.B. Hitchner. 1995. “Roman Africa: An Archaeological Review.” Journal of Roman Studies 85:165-213.

McCormick, M. 2001. Origins of the European Economy: Communications and Commerce, A.D. 300-900. Cambridge: Cambridge University Press.

Glucker, C.A.M. 1987. The City of Gaza in the Roman and Byzantine Periods. BAR-IS 325. Oxford: B.A.R.

Opaiţ, A. 2004. Local and Imported Ceramics in the Roman Province of Scythia (4th-6th Centuries AD): Aspects of Economic Life in the Province of Scythia. BAR-IS 1274. Oxford: Archaeopress.