The question to bear in mind is how the dye penetrates the fibre? Synthetic dyes are usually synthesized under controlled conditions and they will penetrate the fibres evenly with the colour and you even get a deep shade. The natural dyes, on the other hand, are made in the course of time by nature and they have very distinct variations when it comes to the penetration and depth.
Whenever synthetic dyes are exposed to light and constant use, they quickly fade while natural dyes will become softer and their colours become mellow.
When a synthetic dye has faded, the colour it will produce will be entirely different from what it was initially. Natural dyes, however, will not appear different. The colour will still remain but will just be fading out gradually.
You can be able to access the amount of fading when you fold the rug and check the base of the knot. If the rug has a natural dye, you will notice the uniform pattern starting from the tip of the knot down to its base. The synthetic dye will show a very obvious colour change around the tips of the tufts.