When it comes to moving, umziehen can be used for the general meaning of "to move" (e.g. Ich ziehe um "I am moving"), but ziehen always has to be specific - there needs to be a destination.
When you have named a destination, you can say both Ich ziehe nach Berlin um and Ich ziehe nach Berlin - both are equally correct, but each has slightly different stresses.
Using umziehen generally implies that you are moving from one house to another, with all of your things. It can have a more permanent feeling, and stresses the move itself. Ziehen, on the other hand, might be used for a more temporary move, or it could be used to put more emphasis on the place you're moving to rather than on the action of moving.
But these are small differences. In general, if a location is specified, they can be used interchangeably.
I hope that that is helpful!