In some lathes a proper morse taper with the tang will not seat because the tailstock taper inside the quill isn't the proper length. It's too short. With some lathes the proper taper will seat but it will reduce the stoke of the tailstock has because it will eject the taper before getting the quill retracted anywhere close to being flush with the face of the tailstock. In other words you loose the depth you can drill with the tailstock. There might be another reason or two but those come to mind at the moment.
I don't need a short MT2. I was just shopping for a MT2 to try out tbc in this lathe and the Little Machine Shop live center seems to be one that forum members recommend highly.
I wasn't sure what the Little Machine Shop was referring to when they say "short"? If they are referring to the body or the taper being short. The standard taper live center they sell is backordered until April and I was being impatient.
A 'short' MT is physically shorter than a standard MT. As Pete said, there are situations where a short MT is needed, but there is also a downside - lathes that are designed to automatically eject the tailstock MT presume that the MT is the standard length. As a result, the mechanism intended to push the taper out of the quill don't work with short MTs. If the tailstock is hollow, you can always use a knock-out rod to push it out (just like you have to do on the headstock), but if the tailstock is solid, that's not possible. There are a some tricks that can be used to remove a short MT that is stuck in a solid quill; I've never had to deal with the problem, so I've never paid much attention to the solutions, but there are a few videos around that discuss that subject.