What is the connotative difference between and "Lady" and a "Woman"?
"Woman" deals more with gender and gender-identification whereas "Lady" deals more with a list of attributes a woman has.
When I think of the words "lady" and "woman", there are some differences in tone and meaning that help to differentiate them from each other and their use.
For me, woman is a very generic term that indicates someone with either the proper genetic makeup or someone who has the belief that they are indeed a woman (this may be a not-very-elegant way of including trans-women and other self-identifying-as-female people into the discussion, but that is what I'm trying to do. So please know that if there is something in this paragraph that is offensive to you, the intent on my part was to be inclusive, not insensitive).
Let's compare that to lady. There are attributes that a woman must have to be considered a "lady", including grace, elegance, gentleness, sophistication, charisma, and so on. The Disney movie Lady and the Tramp highlights this difference.