If your bed is slipping and the mechanical stuff is OK (pulley tight on motor shaft, belt tight), you're trying to print too fast or you've got the acceleration/jerk set too high. You can still get layer slipping using screw drive. If you try to move too fast the motor will slip.
The main advantage to screws over belts is the elimination of print artifacts like ringing. Screw drive is typically noisier than belt drive because the screws and bearings are pretty efficient at transmitting vibrations from the motor to the rest of the printer.
The main advantage to screws over belts is the elimination of print artifacts like ringing. Screw drive is typically noisier than belt drive because the screws and bearings are pretty efficient at transmitting vibrations from the motor to the rest of the printer.