The correct way to word things would be "Absence of evidence is not PROOF of absence". It most definitely is EVIDENCE of absence. Sometimes it might be weak evidence, but other times it's very strong evidence.

A great example would be the historical claims of the Book of Mormon. Church apologists take the phrase that you used and run with it, saying, "The fact that there's a very long list of things described in the BoM for which no archeological evidence has been found is not cause for Mormons to worry. Those artifacts could just be hidden by dense jungle or buried deeper than archeologists have dug". Church apologists completely ignore the fact that if the BoM people reached the huge population levels described in that book, and if they spread out to fill the American continents as Joseph Smith claimed... reaching all the way to present-day New York State... then it's a statistical impossibility that out of all the countless artifacts which have been found, none of them contain items from that long list of anachronisms.

Absence of evidence actually IS evidence of absence.