This question is quite commonly asked among photographers who are just starting to work with flash photography. Umbrellas and soft boxes are both light modifiers, they are both attached to the light or light stands and light passes through them first in order to get diffused. They come in different shapes and sizes (shape mostly for soft boxes since an umbrella is always obviously an umbrella shape) and the amount of light diffused differs with the shape and size as well.
The key difference between soft boxes and umbrellas are light spill and hot spots.
What is Light-spill?
Simply put, it is the light that goes all over the place. In large studios this isn't that big of a problem, but in smaller ones it can be an issue. You can have problems like more flat light, weird reflections from random reflective surfaces, glare, color cast from walls that aren't neutral and so on. Basically light spill is one of a Strobist's worst nightmares.
Photo by joshuahoffmanphoto
But how does light spill?
Well, in the case of shoot-through umbrellas, part of the light doesn't go through, but instead is reflected back and all over the place. On the other hand, with bounce-back umbrellas, this is less of an issue, but the angles where the light is bounced back still accounts for spill, just generally less of it.
Soft boxes, due to the way they are built, do not spill that much light since the light is better directed and far more controlled. If you want even less light spill, you can use grids to better direct the light (and soften it a bit further as well), or you can use flags made from dark non reflective material to act as a snoot in order to soak in spillage.
Photo by anieto2k
What is a Hot Spot?
Some photographers like them, some don't. I am somewhere in the middle. Hot spots can be good, but they can sometimes be unsatisfactory.
But what exactly is a hot spot? Simply put, it is an uneven distribution of light. The light in the center is brighter and the brightness reduces as it moves further off from the center.
Hotspots are mainly an umbrella issue. Umbrellas aren't flat and have tons of light spill, thus creating hotspots. The center of the projected light is brighter (due to the umbrella center being closer to the light source and flatter) and the edges are dimmer (due to the light being diffracted into the bent angle).
Soft boxes don't have this issue, since they have a flat surface where the light exits the box, and the light is almost perfectly even.
Other differences between umbrellas and soft boxes are the amount of diffusion, size, shape and price.
Photo by Wolfgang Lonien
Umbrellas don't vary in shape that much (minor differences) and the diffusion differs only if you use the umbrella as shoot through or as a bounce back. But other than that, you don't have much control.
Soft boxes on the other hand, can have several layers of sheeting inside that helps with diffusing the light as much as you want to. Also, you can find (and even make) soft boxes in many different sizes and shapes, from tiny boxes for on camera flashes, to 10-meter wide soft boxes for lighting up cars. And their shapes can differ too, from square to rectangle to octagon to hexagon, you name it, you'll find it.
Regarding the price, umbrellas are the cheaper option. Soft boxes (especially the big ones) can be very expensive. But if we are talking about bang-for-buck ratios soft boxes are generally the smarter choice since they last longer. Umbrellas tend to break down more often since they are more fragile (one wire breaks down, and you have issues).
Photo by Wolfgang Lonien
One more difference would be the wind. Umbrellas + wind = bad combination. Even regular winds can make the umbrella tip over a light stand and cause some serious damage. However, soft boxes are a little less dangerous in windy conditions (without saying that it is completely safe to use them).
So, Which One Should You Choose?
It depends on your needs. Umbrellas aren't flash dependant, so you can shoot anything with it, while soft boxes tend to be a bit tricky since they stick onto the flash directly, so you'll need adaptors and that might be a problem for some people if they want cross flash compatibility. It also depends on what and where you shoot, your shooting style and so on.
Photo by anieto2k
Final summation: if you want cross flash compatibility go for umbrellas, but if you want better and variable diffusing go for soft boxes.
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Photography by Gallagher