青白い is a general i-adjective for 'pale' or 'bluish-white', while 血の気がない specifically emphasizes the lack of healthy color from blood, often due to emotion.
顔色が悪い means 'complexion looks bad/unhealthy' and is broader; 血の気がない focuses on paleness from reduced blood flow.
Etymology
Compound of 血 (blood) + の (possessive particle) + 気 (spirit/vigor) + が (subject marker) + ない (not exist). Literally 'there is no blood vigor', used idiomatically for a pale complexion.
