You need a B mandrel if you a using a kit designed around a B mandrel.
Ultimately, the difference between A and B is the diameter of the shaft. An A mandrel is 1/4" in diameter; B is slightly larger.
Some suppliers sell only kits designed around A mandrels, while other suppliers offer a choice of kits that require different mandrel sizes. AFAIK, there is never a choice of A versus B in the same pen kit - its just that some kits are designed for use with larger mandrels.
So that begs the question - why are some kits designed for use with larger mandrels? I suspect that the answer has to do with the fact that some kit manufacturers (in Taiwan or China) design around one size, while others design around the other size, and for no reason other than that's the way they have always done it.
The B mandrel was introduced many years ago by Berea. For a while they made kits that used that "beefier" mandrel, the Churchill and Cambridge among others. The mandrel never caught on, so now you can get "A" size bushings for those kits. So, presently there is no real reason for a B mandrel and I have never heard of a C mandrel.