The first is much more formal. The second is much more common, and less formal.

In a way they both mean the same thing, but yet there is a subtle difference. Acquainted means there was a introduction. It could also imply that you may have heard of her. An old fashioned expression, but still very polite, "I have never had the pleasure of her acquaintance " I don't know her is very direct. You have never met her. You do not know her name. You never saw her before. For example, a policeman could ask "Do you know this woman?" No, I do not. Very direct. Gets to the point.

To have been acquainted with someone means to have met someone. To know someone means you are more familiar with the person beyond the initial meeting there are different degrees of knowing someone from barely knowing someone (which is on a similar level to an aquaintance) to knowing someone really eell, which could be a classmate coworker or friend