It seems that everything here has been pretty well covered. I agree the main difference is between the two is a softbox is far easier to use to control where the light is going. As has been mentioned, you can light the subject, feather the light, scrim it to reduce light on certain areas, keep the light from hitting your background, and move it in closer, probably to the subject. However it is just a tool that was invented to upgrade the umbrella. All these things can be achieved with an umbrella with flags and scrims but it takes more time and space and I'd personally rather be working with my subject.
Another thing to consider when photographing people. Portrait lighting is different from fashion lighting generally, in that in portraiture, you use little or no makeup on the subject. The contours or the face are enhanced by correct placement of the light. In fashion this is usually done with makeup. Therefore fashion people tend to use umbrellas more, set back farther, and allow the models to move around more, which shows the clothes they are modelling to have movement.
When photographing groups, you tend to light more like fashion, with the lights back farther, and create a 3 to 1 ratio. When doing portraits you generally get the lights as close to the subject as you possibly can and then create any lighting ratio you want. You fine tune more, and softboxes are easier.
As for the quality of light between umbrellas and softboxes, they can be about the same. Silver vs white umbrellas are different from each other, and softboxes also have removable baffles inside which will change the quality of the light they put out. Different size softboxes have different effects on the subject as well, besides softness. I have about 5 different softboxes, a mola and a halo light modifier. I like them all, but for different applications. Probably for head and shoulders I like a 24x36 softbox up real close better than anything. I use it with a reflector on the opposite side and a umbrella back behind the camera set about 2 stops lower.
As I mentioned before, doing portraits is not about fiddling with equipment, and tripping over cords and light stands. It is about developing a relationship with the subject, and mucking around with too much equipment will negatively reflect on your photographs.