Authenticity matters to people, relationships and groups. Authenticity is the state of being such that there is no difference between what a thing is, and what it appears to be. This implies a lack of deception, and suggests an absence of misdirection. When people are being authentic, it's like slaking your thirst on a hot day with a long cool drink.
Authenticity dispels illusion between people and allows them to gain traction in communication with each other, which is essential in order to resolve differences, which is in turn an essential component of intimacy. Given that the world is a chaotic place where the difference between awesomeness and fuckwittery is only very slight, therefore being able to distinguish reality is not trivial, and distinguishing the reality of who a person is and where that person stands makes a difference.
Honesty differs from authenticity because I can be both honest and pretentious at the same time. Honesty is a measure of the veracity of the information I communicate, and that is then a subset of authenticity. Authenticity is not just honesty, but it's also a statement of the fact of who I am in respect to that honesty, so authenticity communicates in multiple dimensions. Honesty is boolean.
Authenticity dispels confusion. If you are left feeling confused in a relationship, family, or group, then you might be participating in an interpersonal confusopoly, which profits from promoting uncertainty and insecurity. Interactions in these arenas are almost certainly not authentic, because authenticity confronts those problems. Authenticity is OK to not know the answer, and to be wrong, and to be confused, and so the exercise of authenticity encourages processes that always have the net effect of reducing confusion and disorientation.
Authenticity employs and is exercised via acknowledgement. If authenticity could walk, acknowledgement would be its legs. Acknowledgement is an action you take after you see, understand, perceive, hear, or in some other way receive a communication from someone, having the effect of producing a sense of validation in that person. Acknowledgement produces validation, which assures the other of the realness or accuracy of his or her perception, so acknowledgement promotes clarity and understanding, and dispels illusions and confusion.
Acknowledgement is not always pretty. Acknowledgement applies to failures and successes, bads and goods, critiques and compliments. If in relationship I can't raise a complaint, have my complaint validated, and begin a process of deconstructing the problem in which the other will acknowledge participation, then I am oppressed. If I am oppressed, then I will not be free, and I will be tempted to profit from misdirection in hidden efforts to undermine the oppressor, and authenticity dies. In this sense the death of authenticity, or it's vital presence, is ultimately the responsibility of the whole group.
In a group that is mired in illusion, you should be authentic, but if your authenticity doesn't make a difference, you should quit that group.
Authenticity is the state of being such that there is no difference between what a thing is, and what it appears to be. Be authentic. Demand it from others. Demand that others demand it from you.