Nice post to correct the cabbagery immediately above. Mirror image, such a loaded term 
Minor nitpick - You can in fact discriminate between natural and synthetic molecules, through isotope distributions. This is done in the flavours and fragrance industry for things like jasmine. As you point out, the naturally sourced substance commands a premium in marketing terms.
Coal, natural gas, mines, for the smallest molecules which are then cooked into different molecules and these into different molecules and these… and then these mixed with other molecules, packaged and sent to the pharmacy; also, for items whose atoms don’t end up in the final product but which are necessary in between (mineral acids, catalysts, solvents… sulphuric acid is made from sulphur, which is an “undesirable but economically valuable residue” from refineries, so even though it’s a mineral acid and not an organic substance, it ends up coming from petroleum, gas or coal too; it is used in lots of organic reactions).
Vegetal and animal sources, for other “lego blocks” which are more complicated than the ones extracted from oil, coal or natural gas.
For example, one “inert” compound often found in pills is glucose, which is cheaper to get from vegetable sources than to cook from oil; the active ingredient (say, o-acetyl-salicylic acid) is more likely to have ultimately come from an oil well (and this from ancient trees) than to have been extracted from a plant (and I have purposefully chosen an active ingredient which can be extracted from plants, it just happens to be much more efficient to cook it, with modern technology).
Note that this applies to pretty much anything around you. Plastics. Your clothing. I’m wearing panties which are mostly cotton (a fiber obtained from a plant) with a bit of spandex (a fiber cooked from petroleum), and the dyes could be made from petroleum, obtained from vegetal or animal sources, or a mixture of dyes from several origins.
ETA: the isotope distribution thing was used at least once to detect wines to which alcohol synthesised from petroleum had been added.
You can distinguish them in the laboratory, but the chemistry and biology of a natural and a synthesized compound are essentially indistinguishable. If your compounds behave differently in chemical or biological tests, then they are different compounds.
Chemical compounds isolated from natural sources are also synthesized out of smaller building blocks. Nature is much better than we at synthesizing complex chemicals selectively, but the resulting compounds are the same as those synthesized in a lab. There is no mysterious property of natural compounds that makes them behave differenty that we don’t understand.
I’m repeating the previous answers here, but this is a very important point: The mirror image of a molecule (enantiomer) is a different molecule! They behave differently biologically, chemically and even physically. You can distinguish them easily by observing how they rotate polarized light.
“Natural” vanilla extract is also not a pure solution of vanillin in ethanol/water. Many of the impurities have noticeable (and potentially desirable) flavors. Now I certainly can’t tell the difference, but I knew a guy who could double blind identify vanilla and cinnamon (mostly cinnamaldehyde). Likewise, the active ingredient in Marinol, while the major psychoactive component in cannabis (and not a different enantiomer or diasteriomer) is not the only psychoactive component, so we shouldn’t expect the effects to be identical (never mind that the delivery method is different as well).
I wrote that wrong. He could identify different sources of vanilla and cinnamon.