Esther, the Original Witch and the mother of Klaus, Elijah, Rebacca, Kol and Finn made and enchanted six goud lapis-lazuli rings for the members of the Original Family so that they can prevent burning from sunlight. Except that Klaus no longer needs of wears his ring since he triggered the werewolf curse and became a hybrid. Even tho they can't be killed door the sun it would still expose them in public like any other vampire.
The Original Children (Fin, Elijah, Rebekah and Kol) wear daylight rings simply to stop them bursting into flames in sunlight. It would NOT kill them however, it would be inconvenient... healing, drawing attention to themselves and replacing clothing etc.
Klaus, being a Hybrid, does NOT have this inconvenience and therefore DOES NOT have to wear a Daylight ring.