There are umbrellas, and shoot-through umbrellas. Huge difference.
Shoot-throughs spill light everywhere, 50% bounced straight out of the back and the other 50% across close to 180 degrees from the front. But the light is good and soft and in an average rooom you get a lot of automatic fill-in (whether you want it or not). For instant set-up and go, with unpredicatble subjects like children and so on, in the right situation they give a nice result and lots of flexibility. They're kind of the studio equivalent of a Stofen diffuser cap - really easy and work well - most of the time.
If you want control, you get quite a lot of that from a white or silver brolly, or at least as much as with an average softbox, and no more spill. Slide the head up the shaft a bit.
The advantage with a softbox is the diffuser, preferably a double diffuser, which spreads the light evenly and right to the edges for max softening. You can use them right up close with lovely clean catchlights, round or square
For a lot of regular portrait work, a white brolly is cheap, light, very quick/easy, and will give results indistinguishable from a softbox costing ten times as much. But if you can stretch to a good softbox/octabox, one with a quick up/down mechanism, you'll never regret it.