Your question states that the two halves of the empire found themselves in similar positions, and yet one fell while the other persevered, and tries to find a reason why. The point I was trying to make was that there isn't always a good clear-cut reason, "time and chance happeneth to them all" as the quote says (I remembered it randomly when I was reading your post and then you mentioned Christianity). Why was the British revolution peaceful while the French was bloody? Why did America rebel while Canada remained content? Why did the tech industry develop in Silicon Valley rather than anywhere else? The course of history can be changed by a single decision, by the courage or cowardliness of a single front line soldier, by the sour mood on a certain day of a certain politician, by someone getting sick and missing an important meeting, etc. It's not a very satisfying answer, but it is a fallacy to assume that every historical event can be understood as the direct and reasonable result of measurable factors like population, economic performance, quality/quantity of military armaments, etc.
Dozens (or perhaps hundreds) of books have been written on the fall of the Western Roman Empire analyzing the various factors and causes. It is, after all, one of the most pivotal events in the history of human civilization, probably along with the Bronze Age collapse, the Mongol Invasions, the discovery of the New World, and WWII. The more of them you read, the more you will understand that there is no agreement on a single factor or group of factors.