Silk is fairly warm for the weight, but when talking about single layers of very thin, breathable material, there isn’t going to be much of a noticeable difference, I don’t think. I would think more of the smell factor. Natural fibers don’t build up near as much odor as untreated polyester ones. For self, that would be more the deciding factor. But then again, the polyester will stand up better to washing long term… so various trade offs to consider. Then there is price. Silk tends to be pricey.
You might notice more of a difference in doing something like the following:
Thinnish fishnet, then silk layer, then thinnish fishnet, then silk layer, thinnish fishnet, then silk layer vs say an equivalent system with some polyester or nylon fabric as the convection and conduction limiting factor.
But then again, generally you can get much more tightly woven polyester and nylon fabrics than silk ones, so if you got the CFM levels of each layer just right in the synthetic (say around 80 CFM), it would outperform the silk.
(As a side note, such a system would be an interesting, more truly “warm while wet” than many others, but not all that convenient or cheap! Especially so if each layer of fishnet and woven fabric was made out of polypropylene)