Each damage type is situational.

If you want a weapon that will hit decently against everything, use normal damage. If you want to deal more damage to Fiends and Beasts, use a piercing weapon. If you want to deal more damage to Slimes and Gremlins, use a shadow weapon. If you want to deal more damage to Constructs and Undead, use an elemental weapon.

Keep in mind that some monsters are neutral to one damage type, as they're weak to another and can also resist a specific damage type. Some people like weapons that are good against Vanaduke, and others just like to have variety in their arsenal. Don't let damage types stop you from getting the 5star weapons you've always wanted. But just to be safe, try browsing the Arsenal section of the forums for info.