The the How to play document says
Weapon Attacks. If you’re attacking with a melee weapon, apply your Strength modifier to the damage, and if you’re attacking with a ranged weapon, apply your Dexterity modifier.
Certain weapons and special abilities allow you to apply a different modifier. For example, a finesse weapon is a melee weapon that lets you attack with your Dexterity modifier instead of your Strength modifier.
I noticed that it specifically does not say anything about damage modifier when using finesse. only attack. And in 3.5 it was spelled out that finesse replaces STR with DEX for attack bonus, but not for the damage bonus on the attack.
But I could not find any more specific reference with regard to D&D Next. Therefore:
Does an attack with a finesse weapon apply the DEX or the STR modifier as bonus damage? Assuming 12 STR, 16 DEX, and a 1d6 finesse weapon: Is the damage 1d6+1 or 1d6+3?