Umbrellas are pretty much omni-directional in the direction they are pointed, Soft boxes are much more directional... you can feather the light (use the edge of the light) with a softbox... much more difficult to do with an umbrella. Both can put out good, soft light... softboxes can give more control..
Softboxes give more rapid fall-off into shadow than most umbrellas do. Umbrellas tend to have a slightly more gradual fall-off into the shadow zone. One can feather the light coming from a white reflective umbrella; there is a hotter central spot, and a softer, dimmer "edge" in an umbrella's output.
Umbrellas are almost always less-expensive, and easier to set up than traditional speedring-and-4-rods designed softboxes.
Neither is inherently better or worse than the other. With the advent of low-cost Chinese softboxes sold on the web, softbox prices have had considerable downward pressure; there was a time when even a modest quality softbox was $100 or more, back when $100 was actually "real money".
The single biggest difference in picture outcome has little to do with the type of modifier, and a LOT to do with the photographer.
My experience with umbrellas is that they tend to be a bit more "hot spotty", especially in terms of specular highlights. Umbrellas tend to be good for covering a large area with a graduated, circular light, whereas a softbox is good for focusing on a single subject. Though, a really big umbrella really close to your subject can give similar light to a big softbox.