Are the Quest rewards in True Vault Hunter Mode different from Normal mode or are they just the same weapons from the normal quests but with better stats?
The rewards are identical, but with levels to match. You will also get the same heads/skins from missions. Rewards scale with level after Playthrough 2.5 (after finishing playthrough 2) but only if you take the mission at that level. So if there's an awesome equip you want at level 50, wait until after the end of Playthrough 2.5 to take the quest.
There's no particular difference in playthrough 2 beyond levels and the change in elemental damage multipliers, and the fact that the difficulty (even considering levels/stats) is higher in playthrough 2.