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.

An aquaintance is someone you may know by name, but may not know well. It can also mean someone you do know well, but do not count among your friends because you know them well. To become familiar with someone means you know them pretty well.