Don't pay attention to the above posts. I have been doing research on former Royal titles and styles for years and let me tell you this, all the other posters are completely wrong. This is the reality of the situation..
The "name" by which Sarah is addressed is commonly referred to as the "style for the divorced wives of peers". Since her divorce from the Duke of York, Sarah ceased to be THE Duchess of York for that is the TITLE reserved for the wife of the Duke of York.
Under letters patent issued by Queen Elizabeth II in 1996, no former Royal is entitled to retain the style HRH for that is reserved for a member of the Royal family or someone of the sovereign own choosing, as happened with the then Lieutenant Philip Mountbatten RN who became HRH by will of the sovereign, King George VI (he would never have recieved this rank from his wife automatically and George VI wanted to show recognition of his future position as the consort of the Sovereign).
On the day her divorce became absolute, HRH The Duchess of York became Sarah, Duchess of York. This was a STYLE and NOT a title. It is in effect a name. As the STYLE is effectively a name and NOT a title, Sarah is entitled to no precedence, pre-eminence, position or formal address. Contrary to popular belief, Sarah is NOT a Duchess and stopped being so ever since she stopped being the wife of a Duke. Consequently, she is not entitled to be known as Her Grace, Duchess etc.
So the question is, why is she still calling herself or allowed to call herself a Duchess?
The simplest to explain this is by giving you a scenario. If Mr Bert Bloggins married, his wife becomes Mrs Bert Bloggins (the wife always takes her husbands name in an official capacity in the UK). However when Mrs Bert Bloggins divorces her husband she then becomes Mrs Jane Bloggins as she is no longer Mr Bert Bloggins's wife. This happens as a matter of courtesy to Mrs Bloggins. The change of name in the legal status emphasizes that she is now a divorced woman and although she is referred to as "Mrs", that is a courtesy style, even though she is not married. She is permitted to keep the Bloggins because her name changed legally on the day of her marriage (just like with Sarah Ferguson in fact, who is now legally Sarah Mountbatten-Windsor and not Sarah Ferguson). Therefore the divorced wife of Mr Bert Bloggins changes from being Mrs Bert Bloggins (married with a legal name change) to Mrs Jane Bloggins (divorced but retains the style "Mrs" and her adopted marital name).
To recap, Sarah, Duchess of York means Sarah [formerly] the Duchess of York. The definite article in the name is dropped to address the changed status from Duchess to commonoer. The NAME emphasizes that Sarah used to be a Duchess and is not now. It will become easier to understand (unless Sarah Prince Andrew again) if Sarah ever re-married. Under current law if Sarah married Mr Bert Bloggins, she would then become Mrs Bert Bloggins and lose the "Duchess of York" style. However, if Sarah married His Grace The Duke of X, she would become Her Grace The Duchess of X and still lose the York STYLE.
This happened in the case of Diana's mother Frances. When she married Earl Spencer, she became Countess Spencer. However, when she and Earl Spencer divorced, she became Frances, Countess Spencer without any preceding style and lost any status as the wife of Earl Spencer including precedence. Frances, Countess Spencer then married Peter Shand Kydd and became The Rt Hon Mrs Peter Shand-Kydd (The Rt. Hon being owed to her as the daughter of a Baron). She lost the STYLE "Countess Spencer" forever. The only way that she could ever have revived that title is if she had remarried Earl Spencer.
When Earl Spencer remarried, his wife Raine (formerly the Countess of Dartmouth) became Countess Spencer and remained Countess Spencer until her husband died. When her husband died, she married the so called Comte du Chambrun in the now extinct peerage of France. As a result she became the Comtesse du Chambrun and lost the title Countess Spencer (as she was no longer the wife of Earl Spencer due to his death and her subsequent re-marriage). However, she divorced the Comte Du Chambrun sometime later and wished to revive her right to be Countess Spencer. Unfortunately under British law, she ceased to be Countess Spencer on her remarriage and although she fought the matter in court, she lost. Legally therefore she is Raine, Comtesse du Chambrun as the divorced wife of a peer. However she uses the Countess Spencer style, despite it not being legally hers.
The Royal family following the precedent of the divorced wives of peers. Sarah is NOT (THE) Duchess of York. Addressing her as Sarah, Duchess of York is correct but referring to her as your Grace or Ma'am is incorrect. The only correct way of addressing her is Madam, Sarah or Sarah, Duchess of York. The same applied to Diana, Princess of Wales following her divorce from the Prince of Wales. Contrary to popular belief and verified in Berks peerage, Camilla (as the current wife of the Prince of Wales is THE Princess of Wales but had decided to use one of her husbands subsidiary titles because the TITLE Princess of Wales is more closely assosciated with the late Diana, Princess of Wales.