While I know far more about Azathoth than Giygas, I know enough to say that Giygas is practically Azathoth. You can check out my other post that explain why Azathoth is not omnipotent. I think I'd give this victory to Azathoth. Both Lovecraftian Azathoth and composite Azathoth.
@jwwprod: Besides being described by fans as omnipotent, is there really anything to suggest Azathoth is actually that powerful? It seems like if these Old Gods are really that powerful, they could have busted out of their dimension long ago and conquered the Earth, much in the way Giygas was close to doing.
I used to read a lot of Lovecraft, and besides psychic influences, and the potential devastation if an Old God ever did crossover, they never actually really did anything.
Unless the game has some more to it then what Lovecraft wrote, I wouldn't consider him to be anything like TOAA, or even anywhere close to a tier like that. I don't really know much about CoC though.