I must admit that I keep getting confused about specular light vs diffused light relative to soft light vs hard light. It is especially diffused light and soft light that I mix up, but let me give it a try.
If you see tarmac on a sunny day, the tarmac does not reflect very much light (it tends to absorb the light and get warm for that reason), but as soon as you have rain hitting the tarmac, the water on the surface will start reflecting the light and will do so in a way where the incoming light is reflected pretty close to how it arrived. So all of a sudden the reflections of the light in the water is very bright – I am sure you have tried this driving home late at night; the headlights from the oncoming cars is suddenly a nuisance because the rain set in.
So for me specular light is very much about the “dynamic range” of how the light is reflected on a surface. If the brights are very bright and the darks are very dark, then the light is specular. Remember here that the nature of the light is assessed looking at how it is reflected by the subject.
Diffused light however will have much less contrast between light and dark. It tend to be more “shades of grey” than black and white, to put it simply. The subject appears more evenly lit.
One of the tests that I often do when reviewing a lens is to shoot a piece of tinfoil curled up into a ball. Such a subject will yield specular light where the brights are very bright and the darks are very dark; excellent for revealing issues with chromatic aberrations, which you will often see where the contrast between bright and dark meet.