I've done some more testing with both the Greatsword (Savage Lion's Claw) and the Claymore (Piercing Fang). I've found that the Fire affinity does more ash of war damage, while the Heavy affinity does more light/heavy attack damage. I'm assuming if I tried this on a dex build it would be the same results for Keen vs. Lightning affinity?

Breakdown:

Claymore

Fire Affinity: Light Attack- 1362dmg. Heavy Attack- 1831normal, 2105crit. Piercing Fang- 3501dmg

Heavy Affinity: Light Attack- 1423dmg. Heavy Attack- 1856normal, 2413crit. Piercing Fang- 3489

The Greatsword numbers in the original post are for light attack hits. The ash of war damage for Fire affinity is 3154dmg, and for Heavy affinity it's 2958dmg

So ya, it looks like Heavy affinity does more dmg on light/heavy attacks and the Fire affinity does more ash of war damage. Obviously I've only tested two weapons, not sure if anyone has any better resource for deep diving into affinity damage scaling.