There were several crises in the “late” Roman Empire.
For once, the crisis of the Republic, which resulted in the creation of Principat and later Dominat. This crisis may be related to the fact that due to an end of territorial expansion, the constant stream of slaves ran dry. By then, slavery had by large replaced traditional farmers, who had been the backbone of the Roman army. This crisis resulted in a population drop from about 50 million to about 30 million. (source: Kautsky)
Then, there was the crisis of the third century which almost caused the collapse of the Roman Empire. Ursupator states were formed in the West and the East. This crisis was eventually overcome, and both the Gallic Empire and the Empire of Palmyra were re-integrated in the Roman Empire, but in the Western half, this success did not last for long. By 500, the Western Empire had vanished, while the Eastern Empire remained intact for almost 1000 years.
Now there are several factors in which the Western and the Eastern Empires were different. I believe that in the Eastern empire, slavery never played a role as important as in the Western empire, i.e. there were no huge latifundia with hundreds of slaves, so the end of slavery did not mean that much of a catastrophy in the Eastern half.
Furthermore, the Eastern empire is often said to have been technologically further advanced, had more population and was richer, but I am not sure if or why this would be true.
What are your thoughts?
Edit I cannot help but wonder why most answers claim military circumstances as reason for the decline of the Western Empire. The Roman Empire constantly faught wars with its neighbours, which before never caused to major crises, and the most fericious enemy, i.e. the Persians, were faced by the Eastern Empire, which did not collapse around 500. This wikipedia site lists several theories about the decline of the (Western?) Roman Empire, the vast majority of which do not boil down to military circumstances.