Came across this article online and I thought it's worth sharing.

Yours Sincerely vs Yours Faithfully
Letters often include complimentary closes as part of their basic format. In the choice of a complimentary close, many letter writers are at a loss as to which is appropriate to use: ‘yours sincerely’ or ‘yours faithfully.’

‘Yours sincerely’ must be used when writing to someone that you have met or spoken to. It is necessary that you address them by their first names, sign the letter ‘Yours sincerely,’ and use your first name in the signature.

For example, in the salutation, you can write ‘Dear Miss Emma Watson,’ and in the complimentary close, you can write ‘Yours sincerely, Michael Smith.’

If you are writing to someone but you only know his or her gender, you can address them as Dear Sir or Dear Madam. The correct complimentary close in this case would be ‘yours faithfully’ followed by your initials and surname. In this case, the salutation would read ‘Dear Madam’ and the complimentary close would be ‘Yours faithfully, M. S. Smith.’

If the person that you are writing to is not an acquaintance then use a more formal tone with a formal salutation and the complimentary close, ‘Yours faithfully.’

Although non-business letters are less formal today, it is still necessary to use the correct salutation and complimentary close. I personally use ‘Yours faithfully’ in all business letters, while I use ‘Yours sincerely’ for most of my informal correspondence. Sometimes I also use or ‘Yours truly,’ ‘Yours respectfully,’ or ‘best regards,’ but I always choose my complimentary close carefully and make sure that the salutation matches.

Summary
1. ‘Yours sincerely’ is the complimentary close to use when you know the name and title of the person you are writing to, while ‘Yours faithfully’ is the one to use when you do not know the person’s name.

2. ‘Yours faithfully’ is used in business and other formal letters while ‘Yours sincerely’ is used in friendly letters.

3. ‘Yours faithfully’ is used with the salutation ‘Dear Sir’ or ‘Dear Madam,’ while ‘Yours sincerely’ is used with the salutation ‘Dear Mr/Miss/Mrs/Ms’ followed by the name of the person.

Source http://www.differencebetween.net/language/difference-between-yours-sincerely-and-yours-faithfully/