This is way better than the legacy one since it doesn’t require concentration. 

마법 무기에도 시전할 수 있으면 좋을 거 같은데. 딱히 그렇게 하지 않을 이유도 없잖아

It would be nice to be able to cast it on magic weapons as well, there's no real reason not to.

This is nice combined with the Hunter's Lore feature.  Hunter's lore will tell you if a creature has immunity or resistance to normal weapons, and if so, you can use a BA to make a normal weapon magical.

Does a normal weapon +1 become a magic weapon with +2 after it is casted?

A "+1 weapon" is already magic.  A normal weapon doesn't have a bonus.

From the spell description: 

"You touch a nonmagical weapon. (emphasis mine here)  Until the spell ends, that weapon becomes a magic weapon with a +1 bonus to attack rolls and damage rolls. The spell ends early if you cast it again.

Using a Higher-Level Spell Slot. The bonus increases to +2 with a level 3–5 spell slot. The bonus increases to +3 with a level 6+ spell slot."

So you turn a normal weapon temporarily into a magic weapon, the bonus depends on the level of the spell slot.  But casting it at any level on magic weapon does nothing.  

No concentration, and you can get +2 as a 3rd level spell now - nice!

You could combo this with hirings or skeletons created for the Animate Dead spell. Especially since it works on bows and other ranged weapons. 

Good they removed the concentration requirement.  That kept the spell from being generally useful.  The only use for the 2014 Magic Weapon is if you encounter an enemy that has resistance to nonmagical physical damage (like an incorporeal undead) and your martial characters have no magic weapon.  Getting a Moon-Touched Sword is a cheap way to get access to a magic weapon earlier if your DM doesn't give you a +1 weapon.  Plus a Moon-Touched Sword means you don't need to hold a torch in combat (good if your character has no darkvision), as the sword is your torch.