As far as I can see, using shoot-through umbrellas is slightly cheaper than using softboxes.
Would I give up anything if I went with umbrellas?
Would it make any difference to the images I create?
You have more control over spill and hot spots with a softbox. The hot spots are much less significant with a softbox.
Soft boxes typically have a more focused and sometimes more powerful quality of light while remaining soft. The biggest reason you might want to use a soft box over an umbrella is to control the spill of light.
Where an umbrella will reflect light into a scene as well as transmit through the material, a soft box will force all light to either die or be reflected back through the front to your subject. Whether the result is more power or less power than an umbrella depends on the construction of both if you're comparing light levels.
Using an umbrella, however, can allow you achieve some semblance of fill light for free if you are in the right environment, such as a room with white or light walls and ceiling. The light will not only pass through to the subject, but the umbrella itself becomes a giant ball of light in a room, lighting it up and filling in shadow indirectly. How much fill you get is dependent on the size and colour of the room and walls and the power and type of light used.
One last thing to consider is the shape and quality of the catch lights you want to see in your subject's eyes. Shoot through umbrella produce a typically "round" and solid catch light. Reflective umbrellas can actually produce a non-solid, and sometimes "metallic" looking catch light. Soft boxes will produce a square if you are using a square shaped softbox.
If you want round catch lights but also be able to control spill, you may want to look into octoboxes.
In a nutshell:
Softbox for focused strong but still soft quality of light, with limited spill depending on the construction/modifiers on the soft box (such as a grid).
Umbrella for soft light and to save money where control of spill is not needed or is actually desired.
Catch lights are dependent on the shape and type of modifier.
catch lights: round/octagonal with umbrellas, square with softboxes
ease of setup - umbrellas are generally much easier to set up and attach to a stand
stability - umbrellas tend to catch the wind outdoors and tumble more than soft boxes
spill - umbrellas will spill more light which can be a problem in close quarters (however there are so-called umbrella soft boxes, which are shoot through umbrellas with black backing on them, so are effectively umbrella shaped soft boxes.
feathering - soft box will create a more distinct edge (see link below)
use with IR off-camera flashes (e.g. Nikon CLS) being open, using umbrellas may be more reliable in allowing the camera and flash to maintain line of sight
There is a good post by Zach Arias here with images showing the light spill and falloff of both.