You use "Yours faithfully" if you do not know the name of the person, and are addressing them as Sir or Madam. You use "Yours sincerely" if you are addressing them by their name.
As an ex-Business and Economic teacher to the masses, and having taught 16+ year olds how to right letters (In some cases A, B, C) I can catagorically state that.
Dear Sir / Madam / (Tosser allowed if addressing Bellotti or Archer)
is Faithfully as you do not know the name so you can only place faith that the person will act on your correspondence. You assume you do not know their character.
Dear (Name of any sort)
is Sincerley as you are assumed to know the person so you are being sincere when signing off due to the fact you can be sincere with someone you know.
Hope this helps (p.s. most of my students failed badly)
He got it right - 'faithfully' for someone you are addressing with a generic title, 'sincerely' for someone you are individually and directly addressing. It's not his name, but 'Dear Deputy Prime Minister' is clearly addressing a specific individual.
