The Missouri Compromise was a strategic solution by Henry Clay to keep the slave and free states equal. It prevented conflict between the north and south as both regions had equal representation in congress regarding state senate votes
It was the 13th Amendment
This is a very complex question, but they both collapsed due to:
decentralization and government corruption
plagues/ epidemics
high taxes
and invasions
Some differences are that the Chinese empires saw a revival while Rome didn't.
