Was playing Barbarian Invasion for Rome 1 Total War when it dawned on me, yes the Western Roman Empire was falling to pieces and could easily be seized. Yet, at that very same time the Han Dynasty of China was falling also. Why didn't the Huns just stay there in China and take it over instead of migrating to Europe and sacking Rome?
Also, if someone wouldn't mind explaining to me any difference between the Huns and the Mongols.
Why didn't the Huns just stay there in China
The Huns didn't come from China. We actually do not know exactly where they came from, just that they were living east of the Volga River prior to their migration westwards.
One theory was that it was a climactic shift that desertified their area - a lot of Ammianus Marcellinus' diction frames them in terms of climate and shifting weather, and a recent study points out a major drought as indicated by tree ring data.
Source: McCormick, Michael, et al. "Climate Change during and after the Roman Empire: Reconstructing the Past from Scientific and Historical Evidence,” in Journal of Interdisciplinary History, XLIII:2, Autumn, 2012. Pp. 169-220.