A Self-Tapping Screw can be referred to as simply a Tapping Screw. However, it is also often called a Sheet Metal Screw because they are primarily used in sheet metal.
Whichever name is used, these names are for screws that form mating threads (“tapping” the threads) in a pre-drilled hole in the substrate into which they are driven.
A Self-Drilling Screw is a Self-Tapping Screw with the added feature of a drill point. The drill point looks a lot like the point of a drill. It will drill a hole and form the mating threads in one operation.
Here is where the confusion comes; many times the person specifying a screw will interchange the term Self-Tapping Screw (or “Self-Tapper”) with Self-Drilling Screw. If a screw will drill its own hole it is a Self-Drilling Screw.