One of the differences beteen natural pigments and synthetic pigments is cost. I have no doubt that synthetic pigments have replaced natural pigments solely for this reason--synthetics are cheaper. Natural pigments have pretty much disappeared from the standard paint lines-- you know what those are.
However, Daniel Smith is bringing mineral colors back to the market in their Primatek line. I have very much enjoyed using Primatek paints.
Marigold if you want natural pigments, this is the place to start:
There is also Rublev Natural Pigments, which sells both natural and synthetic pigments:
Closer to you is Kremer Pigmente, one of the very best sources for pigments:
(And then there's my latest methods, finding rocks and pulverizing them into pigments!)