The other day, a photographer asked me when they should use the plastic dome on their speedlight. They said a professional wedding photographer told them to get one to soften the shadows in their portraits. I asked where the professional told them to use it and they said outdoors. It is a common fallacy that domes can be used to soften light from a flash outdoors. I do a lot of work in Balboa Park and I often see professional photographers using a diffusion dome on their flash in open conditions.
To understand how diffusion domes work, you first need to understand how soft light is created. Hard light creates sharp-edged shadows around subjects and on the faces of people. It is something we generally want to avoid in portraits, because it is unflattering to the subject. Soft light is more diffuse and the shadows are much reduced or non-existent. Hard or soft light is created by the size of the light in relation to the subject distance. On a bright sunny day, the light comes from one tiny point, the sun, and the light is hard. On a cloudy day, the light is scattered to become a large light source across the whole sky and shadows are reduced.
Your camera flash works the same way. The closer you are to the subject the softer the light will be, because the light source is larger in relation to the subject. As you move away, the light source gets harder and shadows become more pronounced. Of course, a flash is a pretty small light source anyway, so varying the subject distance doesn't make a noticeable difference in most cases. Putting a cloudy screen over your flash does absolutely nothing to soften the light, unless the screen is bigger than the flash and is therefore making the light source bigger. This is how soft boxes work in the studio. They spread the light from the small strobe head over a larger surface area, therefore making the source bigger and softer. The closer you use the soft box to the subject, the softer the light will be.
On a speedlight, we have the option of directing the light straight at the subject or turning the flash head to bounce the light off a wall or ceiling. If we bounce the light, it spreads across the surface and reflects back at the subject, making the light source bigger and therefore softer. If we point the flash straight up, the light from the flash covers much of the ceiling and is reflected back down on to the subject as a much softer light. The issue with doing this is that all the light is coming from above, which can result in dark circles under the eyes called "raccoon eyes" on the subject. You can also see a dark shadow under the chin.
Using the dome diffuser sends some of the light straight to the subject while the rest is scattered upwards and around 360 degrees, greatly softening the light as it is reflected all around the room. You will also see a catch light in the subject's eye as some of the light goes directly forward and back to the camera. Therefore, when I use a dome, I point my flash straight up at the ceiling.
When you soften light by making the source of light bigger, you weaken it considerably, so the bigger your soft box or further you spread the light the more power you use. So, if you use a dome outdoors, you are doing nothing to soften the light, as you are not making the light source appreciably bigger. You are scattering the light, but there is nothing for the light to bounce off back to the subject, so much of the flash power is wasted. You are creating the same effect in your pictures as not using a dome, but probably using at least four times the battery power to achieve it. You can experiment with this by taking pictures outdoors with and without the diffusion dome. As long as you light the subject the same to compensate for the loss of power when using a dome, you will not see any difference in the pictures.