Diffusing the situation
Time for my yearly update on the topic of diffusion and how it affects your lighting. As usual, let's start with a definition of diffusion:
• the spreading of something more widely
• the action of spreading the light from a light source evenly so as to reduce glare and harsh shadows.
I will start of by taking some exception with the second definition. The first part about the spreading of light is correct. Reducing glare earns a "maybe," but changing the angle of the light will be more effective (remember the angle of incidence equals the angle of reflection). Now about those shadows... diffusion on its own (and I am talking here about putting some diffusion material right on the light, not using an umbrella or softbox or bouncing off a wall) doesn't reduce shadows. It doesn't soften shadows. It hardly does anything to the shadows except make them less dense (lighter) by bouncing light off of more surrounding areas (walls, floor, ceiling). It mostly reduces the output of the light as it makes the light spread out wider to cover a larger area. Diffusion on the light is used to allow the light to cover a wider area when using a wider angle lens. Or it might be used to even out the spread of light across the surface of the light (useful for evenly lighting a background, for instance).
Shadow edges, which we use to describe the quality of light, are determined by the size of the light in relation to the subject. Adding a sheet of diffusion material to the front of a light does not change its size, and doesn't change the shadow transition. The diffusion will make the light disperse in a wider pattern which makes the light seem less powerful as it covers a wider area (which might be a concern for battery-powered lights). It will even the pattern of illumination to make the light more uniform on the subject (more noticeable with larger light sources). But it won't make the shadows any softer. To do that you need to make the light bigger. Do that by using a scrim that is placed a distance in front of the light and that is much larger than the light.
Here is a set of images lit with a 7-inch dish reflector and with a 20-inch dish reflector. The first image in each series is the bare reflector. The middle image adds one sheet of diffusion material over the reflector. The third image has two sheets of diffusion material on the front of the light. Look at the nose shadow in each as you go across.
The shadow edge across each row does not change as diffusion is added. To change the shadow edge and make it softer you need to make the light larger. Notice that the 20-inch reflector with no diffusion is softer than the 7-inch reflector with double diffusion. Diffusion material on the light spreads the light to cover a wider area, it doesn't soften the light. No matter how close you bring the 7-inch reflector to your subject, or how much diffusion material you put right on the reflector that light is never going to be larger than your subject's head, and hence will always be a small/hard light source. Soft light comes from light being able to reach your subject from a variety of angles instead of just straight on. Look at the diagram here. You see that from the small light source all the rays are concentrated leading to hard shadows. A bigger source from the same position provides its own fill light to soften the shadow. Some people say that the light "wraps" around the subject, but it is all traveling in straight lines. If you were to take that larger light source, though, and back it up a ways it will become relatively smaller compared to your subject, and hence more directional causing harder shadow edges. So what about the scrim I mentioned above?
Make it bigger!
Yes, you can make the light bigger by using a diffusion scrim, but the scrim has to be separated from the lamp, not used right up against it. Here we have the same 7-inch dish at 36-inches from the face with "double diffusion" and next to it the dish with a diffusion scrim added about 17-inches from the face. Notice the big difference in the shadow edges. The strobe head (the origin of the light) remained in the same place. But the source of the light changed from the strobe head (small) to the scrim (large). The scrim made the light larger and closer, both of which help soften the light.
This is where I think the confusion comes in. In both cases the light has been diffused by translucent material. The difference is in where the diffusion is applied and how it affects the size of the light and thereby the quality of the light.
It comes down to this... Use diffusion to spread the light wider, use size to control the quality of the light. A large undiffused light is going to be softer than a small diffused light.
There can be an advantage to using a small light with a scrim over using an umbrella or a softbox--more control. You can place the scrim at the position you want to control the falloff of light from the subject to the background and then you can move the light in closer to the scrim making it smaller to make it harder or back it away to light up more of the scrim making the light softer.