The King
1. His splash magic attack is healing. The amount he heals is determined by your magic. Take your total magic (Stats + Weapon + Pet) and divide it by 2. If the amount ends in a decimal, round it up. I'm not sure if it heals the party in co-op but I'm assuming it does.
2. His RT+B and RT+A+B throws golden knives.
3. His magic aura is golden.
4. He starts off with the King's Mace. (It's the official name for it, not the King's Scepter.)
5. I believe, but I need more opinions on this, that the King causes enemies to drop more money. I noted this during the beginning when I saw the Barbarians sometimes dropped more than a 5 gold bag. I don't know if that's normal but I saw them often drop more than usual.
6. So far (I just beat the Corn boss), even though you're the King, the scenes involving the King don't change. Such as the beginning with the Wizard stealing the crystal, the boat during the Catfish boss, and the King riding his horse during the Flowery Fields. The only change I seen so far was that the King NPC wasn't at the beginning of the Tall Grass Fields.
Open Faced Grey/Gray Knight:
I've read from other posts that the Open Faced Gray Knight has no new abilities, he is exactly like the regular Gray Knight, just that his visor is open.
Pelter the Seal
The new pet is a seal named Pelter. He occasionally shoots ice. I tried him and he's not very useful. I don't know if he freezes enemies or not, but I assume he does.
Weapons
NG Lollipop - Level 1 - +5 Agility
NG Gold Sword - Level 1 - +6 Strength, +1 Magic, +1 Defense (Awesome!)
King's Mace - Level 20 - +3 Strength, +2 Magic, +3 Defense
Extra
As for beating the game with either character, I have yet to beat the game with them, but I doubt anything new will unlock. If you beat the game with one of these characters please feel free to post with what happened. If I made any mistakes, feel free to correct me.