In "Japanese for Busy People I" page 2 (kana or romaji edition) it says that いいえ is 'virtually the same as "no"'.
It sounds like there are some differences, but they want to omit mentioning them for the time being. What differences are there?
The key to whether to reply with はい or いいえ is not whether the form of the question is positive or negative, but whether the question is asked in a way which expects a positive or a negative answer:
一緒に行きませんか？(expects a positive answer)
一緒に行かないんですか？(expects a negative answer)
I guess this response could be regarded as cyclic, since the obvious next question would be when a question expects a positive or a negative answer. Usually positive forms expect positive answers and negative forms expect negative answers. The tricky ones are the simple negative-form+か (without んです or のです), which are often used for invitations, or to ask for agreement. An example of an invitation is given above. Below is an example of asking for agreement.
Returning to いいえ, I would say that in practise, it's mostly used in the sense which gibbon describes. Using it as "No" sounds pretty strong. More often, the main verb/adjective is repeated in positive/negative as needed.
They might be refering to the fact that いいえ can also be used to essentially say "that's ok" / "no problem" / "don't mention it" when someone appologizes to you or thanks you for something.
Or that there's other ways to say "no" and that you shouldn't take いいえ as the only correct way to say it in all circumstances.
As a matter of etiquette it's perhaps also worth mentioning that 「いいえ」is not appropriate in all situations where "no" might be appropriate in English and many other languages. It's important to keep this in mind in real life conversation.
For example, when turning down an invitation, 「いいえ」 would often be considered rude, where "no thanks" may be fine in English.