If you have both, you'd use both. Brynhildr's heal is better as a primary source of healing, while Amaterasu's works great as a secondary. Outside of healing, they don't overlap in what they offer. And healing is one of the things that you can get away with doubling up on.

In a vacuum, we can get a rough idea of a defense buff's value from looking at the damage formula, but combo attack buffs/debuffs have not been nailed down. Wiki cites Amaterasu's def buff as A-frame +30%. That's really nice, when defense's involvement in the damage formula looks to be (product of attack stuff multiplied together) / (defense +/- buffs and debuffs). If there are no other modifiers to defense involved, throwing on that +30% changes damage from (attack_product)/1 to (attack_product)/1.3, which is basically reducing damage by a quarter. A nice, big defense buff is one of the handy tools you have to reduce boss attacks from whatever gaudy numbers they originally have to far more manageable levels.