They used to be the same word in English, too, until the spellings diverged. They both derive from the Latin conscio, which means to know or be conscious of (something).
In one sense, the thing that is known is the self; in the other, the thing that is known is an interior awareness of right and wrong. Philosophically, you can see how those two things could have been considered very closely related.
Though according to the Real Academia Espanola, "consciencia" is an alternate spelling of conciencia AND it defines it as means consciousness (awareness of self). It seems the spellings just haven't diverged as far apart as they have in English.