Umbrellas cost less, size-for-size, and are typically easier to set up and transport. Umbrellas are self-contained, and you never lose or forget the speed ring or the rods of a softbox--the umbrella is entirely its "own entity" if it is used with a studio flash, which has the umbrella receptacle built in. When using speedlights, you do need to always remember the umbrella swivel mount.
Umbrellas come in many styles and colors--white refelctive, white shoot-through, gold reflective, silver reflective, convertible, and deep parabolic like the new Paul C. Buff Parabolic Light Modifiers or "PLM" umbrellas which are **SUPER-EFFICIENT** and can light up incredible areas at long distances, with much,much more efficiency than any umbrella I have ever seen, yet which are affordable. Speedotrons Super Silver style silver metallized umbrellas give crisp,beautiful lighting that looks fantastic in B&W.
Soft boxes from Chimera and Photoflex are high-quality and last a long,long time, but cost more than cheaper brands. There are many CHinese-made boxes sold via
eBay for very low,low prices, like $29 to $59; I bought two Chinese cheapies, both smaller 24 inch models, one with a grid and recessed face, the other just recessed face, for only $35 each! And they both do surprisingly good lighting, although the speed rings and the assembly rods are a biatch to work with compared with the quality, better-engineered Photoflex boxes I've used for 20+ years now.
There is also a wonderful hybrid umbrella/softbox: the Lastolite Umbrella Box, and the Phototeck Softlighter. These are enclosed, reflevting umbrellas with a diffusion face that zips or drawstrings closed, and they have some of the best qualities of both umbrellas and softboxes. Annie Liebovitz loves the Softlighter. I love the Umbrella Box, which you can see a lot of examples from by stopping by the dg28.com web blog, which is Neil Turner's superb resource for on-location lighting using almost exclusively the Lastolite Umbrella Box and Lumedyne battery-powered strobes. I believe that Turner's blog is/was the inspiration for the Strobist blogspot.