What effects do different flash diffuser shapes have?

The point of a diffuser is to create light that comes from a larger area and is more spread out than what the bare flash produces. Theoretically, its shape influences the distribution of the light as well, but due to the spreading-out effect, light basically goes from every point on the diffuser's surface in all directions. Which means that in practice, a square vs. an octagon will not make a noticable difference (except in highlights, where the subject directly reflects the light source). Especially since the corners of the square one will get less light from the flash in the first place.

What matters most is the size of the diffuser, its distance from the flash, and its practical usability - things like the mechanism for affixing it to the flash.

The primary difference in diffuser shapes is in the shape of any specular reflections in the image, for example the catchlights in the subjects eyes.

Octagonal diffusers produce highlights which look more rounded and therefore more natural and less obtrusive.

However, in some cases hard edged highlights are required for a certain look, you see this a lot with product photography, particularly that of the Apple corporation!

_{(source: wikimedia.org)}

"IPhone 5C" by Justin14 - Own work. Licensed under CC BY-SA 3.0 via Wikimedia Commons.

The difference the diffuser shape makes is mainly the shape of the catchlights.

I.e - shoot a person with a square diffuser, you'll get a square catchlight in their eye. Shoot them with something round, the catchlight will be round, which arguably may look more natural. On the other hand, square catchlight can look like a window, which is also neat.

There's a section in this article on Adorama that shows the difference pretty well, specifically with this image:

On a general note about diffusing, those small on-flash diffusers don't do too much to diffuse though unfortunately. Of course it's better than nothing but you won't get anywhere near soft light if you're shooting people. Shooting small objects though it can be useful and actually better than a very large diffuser, since that'll soften the light too much and almost erase any visible shadows for a small object.