I am having a really hard time thinking of an occasion where they could not be used interchangeably. First of all, I think they are both usually reflexive; in other words, you acquaint/familiarize *yourself* with something. This is not a perfect rule, though; you can say "The teacher familiarized the students with the finer points of grammar." I guess I am just thinking about my own usage. One important difference is that the past participle 'acquainted' has much more widespread use than 'familiarized.' I would say that "acquainted" is a state (the people are acquainted; I am acquainted with C++) but "familiarized" is just a true past participle, so the result of having been familiarized. Whereas you can say that you are acquainted with something or someone (meaning you are in the state of knowing it/them well), instead of saying that you are familiarized with something, you say you are familiar with it. So acquainted = familiar in this sense. I hope this helps. This is a great question about a nuance I had never thought about before.

Hi Baron You are right they both mean to "know" someone but Acquaintance is usually someone you don't know that well. Eg, I met an old work Acquaintance the other day. Familiarity refers to someone you know really well from the root "family" There is a saying in English that helps to explain it. "Familiarity breeds contempt" meaning if you know each other so well you could start disliking each other. I hope this helps