Hi Sixpence,
A ball screw and nut assembly has a circuit which is filled with steel balls. As the screw rotates, the balls roll in the "threads" of the screw and nut. This provides the low friction of the unit. Because the balls would just fall out eventually, the ball nut has a return tube ciruit that scrapes the balls out of the groove in the screw and returns them back to the beginning of the ball circuit at the front end of the ball nut. The thrust action of the screw is non-wearing because the balls roll, rather than slide.
A lead screw is a simple bolt and nut, with conventional V threads or Acme threads. The screw threads slide past one another, with considerable friction and subsequent wear. They also require quite a bit of free play to ensure that lubricant can enter, or else there is a danger of the threads seizing up.
You cannot easily adjust the backlash in a low precision ballscrew, if it has a single nut construction. If it has double nut construction, then the two nuts can be shimmed apart and then locked in place, to reduce the backlash.
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