Why Use a Softbox or Umbrella

When we use an auxiliary light on our subjects we create shadows. Although these shadows may enhance or bring out texture, they may be too distinct and harsh, much like bright sunlight. Light on a cloudy day produces softer, less distinct shadows giving our photographs a softer look. This is particularly welcomed on most portraits.
A softbox allows us to aim the light through a diffusion material while bouncing the light around the inside of the box to produce more diffusion. The result is a much more even distribution of light which produces a softer image, much like a cloudy day.
The umbrella works in a similar fashion my directing the light to the back of the umbrella, bouncing the light, causing the light to scatter and diffuse creating a softer light on our subject.
Although both can sometimes create a flat photograph if used straight on, by moving the softbox or umbrella to the side more shadows can be produced to give more shape to our subject although less so with the umbrella. These shadows will be less distinct that that from a bare light. The use of a white reflector will complement the soft light of these two modifiers.
The softbox will give a more directional, but diffused light, while an umbrella will produce a more diffused and non directional light that tends to wrap around the subject, particularly at close distances. The softbox produces less light spill while the umbrella is more difficult to control because of its large amount of spill light. The softbox will produce more pronounced shadows yet a soft light.
Either one can be used as main, fill or other even a hair light. Just consider the limitations and strengths of each one.