Authenticity suggests that we are what we proclaim to be. It implies a genuineness about us. Those are good things. Authenticity, however, is subject to perception, to image management, to illusion. It is not at all difficult to appear authentic, yet not be. This machination is successfully deployed rather frequently by politicians, con artists, salesmen, and leaders of all stripes.
Integrity, on the other hand, goes beyond suggestion and implication. It is the proof in the pudding. Integrity is the consistent manifestation of trustworthiness, the unwavering alignment of our actions with our words. Integrity is the dependable and consistent enactment of our espoused beliefs, values, and tenets.
Authenticity is about depiction. Integrity is about conviction.
Authenticity can be faked. Integrity can't.
Integrity trumps authenticity.