In the old days it was considered rather crude to call someone a "woman" unless it was meant in a bad way- implying that she was "no lady".
Many people of the past generation still automatically use "lady" rather than "woman" out of respect.

You know, the US is a big country full of people of all ages and regions and cultural backgrounds. Terms like lady, woman, Miss, Ma'am, etc are natural regional, generational, and cultural variations- perhaps we should not get too wrapped up in trying to get everyone to use the terms we prefer. I do agree that calling women "girls" is rather dated and at this point.
I lived in Puerto Rico for many years, where I learned to refer to a boy child using a certain term. I was talking to some Mexican women the other day in Spaish, and when I asked if one of them had just given birth to a girl or a boy, they looked at each other all shocked and then burst out laughing....what had I done??? Well apparently in Mexico one only uses that term to describe a male ANIMAL, never a person! So hard to be correct with everyone! Too funny.