Lady and woman are two titles that are used to refer to a grown-up person of the female gender. In many societies, both words are used interchangeably, depending on what comes to your mind if you want to use it for someone. Most times, the word 'lady' is used to refer to a woman who has certain qualities like a good sense of dress, good character and manners, etc.
At times, the word lady is used as a title for women that are in a higher position or class. It is believed that anybody could be referred to as a woman, but the moment the word lady is used, it has set a major difference.
On the other hand, the word woman is the opposite word for men; it is used mostly for older people of the female gender. For instance, you can use a woman to refer to someone who is above 40 years of age, but it will be inappropriate if you decide to use the same word for someone who just clocked 22, but 'lady' could be an appropriate word.