Daenerys appears to be protected though an immunity (or at least, a high resistance) to fire, whereas the Night King appears to be protected through the sheer cold that radiates from him.
While it isn't depicted consistently, fire in the vicinity of the Night King does tend to snuff out, as seen in S06E05 "The Door" when the Children of the Forest create a firewall to defend the Three-Eyed Raven and Bran from the Night King, who simply walks through the fire and snuffs it out.
Put in a more 'computer gamey' way, Daenerys's fire resistance ends at her skin, whereas the White Walker's cold is more like an aura effect.