For me the main difference is a softbox generally has better control of the direction of the light. The softbox will have a flat diffusion panel on the front and possibly a raised edge that stops light spilling off to the side. You can add a grid to it to control the spill even more. While an umbrella, has a curved surface that reflects or diffuses light in a more uncontrolled way. If you were in a small room using a shoot through umbrella you'd get light spreading all over the place as well as where you were actually pointing it. This isn't necessarily a bad thing.
The softness/quality of light are so subjective but softboxes can have multiple diffusion panels so you might get a more even light spread. Softness if going to depend on relative size of source and subject, and contrast is going to be determined by your light source to subject distance. A silver reflective umbrella will probably give you higher contrast/more specular highlights.
Umbrellas are simple to put up and down and can be purchased very cheaply. They just mount to your light/light-stand with their pole so they are universal. Soft boxes usually have a specific mount and can be a pain to assemble.
There are brolly boxes/easy up softboxes which open like an umbrella but are enclosed and have a flat front of a softbox.
In short:
Umbrellas: cheap, easy to use, versatile, less spill control, reflective umbrellas are difficult to get close to subjects because of the pole.
Soft boxes: generally more expensive, more spill control, better for larger sizes, more even light spread, need to get them with the correct mount to match your lights
