How did the Compromise of 1850 resolve the differences between the north and the south?

It didn't.

The problem was that it was getting more difficult to create new slave-states, so most of the new territories were probably going to become free states of the USA.

To stop the South from breaking away, Congress had to make a big gesture of appeasement. So they settled for a heavy crackdown on runaways - the Fugitive Slave Act.

This backfired badly. Northern citizens strongly resented being treated as unpaid slave-catchers, and 'Uncle Tom's Cabin' was written as a protest against the Act.

The Compromise of 1850 certainly didn't resolve the differences between North and South.