Not exactly. GMO foods are genetically different by definition. Whether or not the difference is significant is another question, but natural vs "synthetic" diamond really is closer to the question of where a plant is grown, or perhaps if it's wild vs. planted.

So it's more like the Champagne versus Sparkling Wine situation.

Right, it's only a Diamond if it's from the bloody regions of European colonialism, otherwise it is just Sparkling Carbon.