Octoboxes and more-typical rectangular softboxes appear to have similar structure and function. How do they differ, and when would you use one over the other?
An octabox will give you nice round catchlights and produce generally more natural looking highlights and reflections. The straight edge of a softbox often sticks out when shooting with reflective surfaces more than a more organic curve or circle.
On the other hand softboxes are easier to mask and gobo due to the straight edges, and more suitable to certain technical lighting styles (e.g. for product photography where you want lights parallel to certain surfaces or at an angle to 'feather' the light). They are often cheaper, and easier to set up and tear down.
You can't go too far wrong with either, I prefer boxes in small sizes for simplicity / control and octas in large sizes for a natural light look and inconspicuous reflections.
Octaboxes tend to focus the light a bit more than square or rectangular softboxes. Some fashion and portrait photographers favor them for several reasons:
In terms of the "more light" point I brought up, the fact is you can only get so many lumens out of a give light at a given power. However, the diffusion of that light absorbs some of the light, reducing the amount that reaches the subject. The more narrowly the modifier focuses (think parabolic reflector), the more light appears to fall on the subject. In this regard, my opinion is that octabanks are very useful.
You'll also find some really big octabanks that are useful for lighting wide or tall scenes evenly.
I would typically use a softbox for less-than-perfect complexions, older (i.e. > 35 years) people, and products where shadows are to be minimized. Octabank is a mod I favor for younger models who have good complexion. I don't use them for product.