For cutting threads with Taps or Dies, the holders are a body which slides on an arbor with a Morse taper which matches the Tailstock.
In this way, the newly cut thread has only to drag the holder along the arbor, rather than the Tailstock along the bed. (A fine thread is likely to strip when subjected to the forces needed to drag the Tailstock along the lathe bed)
Also, if tapping a blind hole or cutting a thread upto a shoulder, a mandrel handle allows chuck rotation to be stopped as soon as any resistance is felt, preventing broken taps, or stripped threads.
With regard to setting split dies; wherever possible, I cheat by adjusting the Dies so that a commercial bolt gives a nice gentle feel (Unscientific!).
The other advantage of holding a tap in an ER collet, or a drill chuck, (Sliding on the arbor) is that often the tap slips before it breaks.
If the holders are a slightly loose fit on the arbor, the clearance allows the Tap or Die to self align on the workpiece, minimising the risk of a drunken thread.
Howard