This is not a solid rule. But your question asks "what is more common..." If I were talking about a thing, such as a motor, computer, or information in a book, I would expect the word familiar to be the first choice between these.
If I were talking about a person, acquaintance may be expected out of these two (though it is used rarely). I am suggesting that acquainted holds a relationship connotation that would not be applied to "items" very often (where as familiar can apply to an object, animal, idea, or any person that you know at almost any level.) The word acquaintance (or to become acquainted) is most often used when being introduced to someone for the first time.