Here are my (somewhat lengthy) thoughts on the matter...
I believe the basic material composition is the same,...that is, all of it is "latex". "Proph" stands for "prophylactic", which is basically the very thin latex material used for certain products such as condoms. In terms of turkey call construction, we use the term proph to describe that material,...and yes, it is generally thought to be .0025 in thickness (or thereabouts). Natural proph material is the white/clear material we generally speak of, but there are other proph colors available.
Latex is made in batches of large sheets and is made for many applications. The functions of the coloration of each batch is both decorative and for identification of thickness of material for use/application by the industries that utilize it. However, the same thickness of material is often made in many different colors to give consumers options in that regard. Obviously, latex sheets are not produced specifically for use in making turkey calls, so we are basically taking a product used for other purposes and applying it to our craft. Because of that, there are a lot of "ifs" involved when we take a particular piece of latex and apply it to making calls.
There is debate among us mouth call builders at to whether the color added to latex actually affects the sound quality of the material as compared to the final material thickness produced in the "batch" of latex it comes from. There is no doubt that a particular batch of material will have an impact on the sound quality produced in a call, but whether or not that is a function of the color added to the batch, or is more related to the very small differences in the thickness/composition of the overall batch of material is questionable.
As to your question regarding the "advantages" of the different thicknesses of material, thinner material (proph, .003., .004) is generally easier for most callers to use. However, the thinner the material used in a call, the more fragile it becomes in terms of maintaining its sound quality over time. In addition, there are many other factors that determine sound production in any given call, not to mention the abilities of the caller.
We could get into more detail regarding those factors if desired, but I'll stop for now.