All on here so far have been in favour of softbox because of added control, and I am fully aware and agree to an extent, but for the benefit of the OP (nothing to do with the fact I'm rather bored), I will play the devil's advocate here, and fight for the brolly corner!!!
Shoot through brollies are cheap, easy to transport and position close to subject. Catchlights are also round, as opposed to (most) softboxes which are square/rectangular. I find round catchlights to be more pleasing.
When you turn up at a client's house for a portrait sitting, do you want to make your client wait for 10 min while you set up your softboxes, or spend 30 seconds putting 2 brollies up, and get shooting?
Yes, it does spread light everywhere, so if you work in a small space, and or you want to creat a low key, selectivly lit portrait, then the spill will be difficult to control. However, if your primary aim is to create well, evenly illuminated family portraits, then a white shoot through brolly will do a very fine job at that.
As for the brolly box thing, just the same as a reflective, you can't place it as closely to the subject as you can with a shoot through. So with the same dimension, the relative size of the light will be smaller (hence harder) than a shoot through. And while I have no experience, I'd imagine as they have no front recess, the directional control is hardly greater than that of a shoot through brolly.
Control is a great thing, but in terms of convenience, you just can't beat a good ol' brollie. So there's a couple of reasons for using them in place of softbox.
Counter arguments welcome