Generally speaking, you can't push with a rope.
Tie a rope to a doorknob. From across the room, you can pull the door open, but you cannot push the door closed; the rope just goes slack.
Ropes are only useful when they're in tension.
Re-read your question and realize I didn't answer it all all.
Suppose you have a hinge joint. Use the door example again. The cable actuators attach to either side of the door.
As the door rotates, each side scribes an arc, but because the door has a thickness, and because the hinge is on one side of the door, those arcs with have different arc lengths.
Kind of hard to describe without a picture (on mobile), but side one scribes $r\theta$, where $r$ is the distance between hinge center and the hinge side of the door, and the other side scribes $(r+d)\theta$, where $d$ is the thickness of the door. $\theta$ is how far open the door is.
Hopefully you can see that, if you wrapped a string from one side around a drum and back, the drum would have to "take up" a quantity of rope $r\theta$ while it tries to "pay out" a quantity $(r+d)\theta$, or vice-versa. This is not possible with a regular drum.
Welcome to the frustrating world of web tension control.