I've never been that interested in gemstones of any sort, but my understanding (from geology classes decades ago) is that natural diamonds on the surface of the earth result from some rather exotic processes.
First, they form deep down in the mantle around 100 miles down. The only way they get up here is through rare and violent high velocity eruption events that bring mantle material up to the surface. Most are very old, too; greater than a billion years old.
I don't recall the source of the carbon they're made from, but it's certainly not any sort of organic material like coal or dinosaurs. ;)
Or I could be totally wrong. I really have no interest in these things beyond the geology behind them and it's possible the science has uncovered new stuff since I last studied them.
Likewise, I have no idea how these synthetic diamonds are being made nowadays, but I bet that's pretty cool, too.