I'm not sure why you say 'remained' is passive. It is an intransitive verb so I don't think it can ever be used in the passive voice.
His eyes kept closed. He kept his eyes closed.
His eyes remained closed. He remained his eyes closed.
The main difference between the verbs is that 'to remain' is always intransitive and 'to keep' is transitive. Even in a sentence like "Keep quiet!" the direct object is implied (Keep yourself quiet). In the sentence "Remain quiet!" it would be incorrect to include 'yourself'.