When making a full-attack action with a bow, your first attack fires two arrows. If the attack hits, both arrows hit.
Once per round when you would normally be hit with an attack from a ranged weapon, you may deflect it so that you take no damage from it.
If you use Deflect Arrows to stop the first attack of a Manyshot you take no damage.
Manyshot is a single attack. Deflect Arrows negates the damage from a single attack. Deflect Arrows wins.
Corollary:
Does Snatch Arrows do double damage if used on a Manyshot attack?
Snatch Arrows doesn't do any damage at all. It does allow you to immediately make an attack with a thrown weapon when you snatch it, but that doesn't apply to arrows and the attack and damage is based entirely upon the person who is now throwing (the guy who snatched the weapon) not the original shooter and would use that persons feats etc.
If we imagine a theoretical manythrow that let you throw two daggers as a single attack, the snatcher would probably (free hands might be an issue here) be able to catch both, but they would only be able to throw a single dagger back.