The East Roman empire faced many of the same problems as Western portion.
However, it had 2 main advantages that helped it survive.
Firstly, Constantinople was positioned at the crossroads of East and West, so was a major trading centre for both.This meant the Eastern Empire was much richer than the West, and lots of money came directly to Constantinople (and therefore the emperor) because of this trade.
Secondly, geographically Constantinople was much easier to defend than Rome. Any successful attack required a large, powerful warfleet, something barbarian armies didn't possess.
Military reforms in the early 5th century by the emperors Zeno and Anastasius halted the army decline that was weakening the West, but essentially the Eastern empire couldn't spare much in the way of resources to help the West, concentrating on its own survival.
As for the relationship between the 2, it was more often one of mistrust and interference rather than close alliance and friendship.