I would say the defining characteristic of any light source - diffuser, light shaper, brolly, beauty dish, softbox, call it what you like - is size. Bigger gives softer shadows. To take the ultimate extremes, bright sunlight casts hard, dark, sharp-edged shadows because the sun is relatively very small. On an overcast day when the light source is effectively an entire cloudy sky, there are virtually no shadows at all.
Shoot through brollies tend to be quite big, and blast a load of light to the sides and behind them which is then reflected back from the walls/ceiling etc. This is completely uncontrolled and the final effect varies according to the room. Softboxes are similar, but catch all the waste light spilled around randomly so is much more controllable (and efficient) and can also be used close with a nice clean highlight in the eyes. Lovely
Reflective brollies are much more controlled than shoot-through, very cheap and easy to use. I like them a lot, but because of the way they mount you can't get them very close to the subject and the flash head sits in the middle, casting a slight shadow and making ugly highlights in eyes (IMHO). A large silver brollie used at a little distance gives a similar effect to a smaller beauty dish used close, but because of the distance the fall-off is less which maybe means it is less controllable? And the way you tend to use a beauty dish makes a brollie unsuitable a lot of the time. Subjective thing.
Basically, bigger is softer, and the various reflectors and diffusers are designed to put the right amount of light where you want it, evenly spread, and to control where you don't want it (spill). Within those very basic generalisations, the differences tend to be relatively subtle.
IMHO
