An acquaintance from many years ago, a fellow Beatle-maniac, was once describing life in Russia to me. He said it was like a beer commercial popular at the time that posed the rhetorical question, “Why ask why?”. This happened to be just days before the breakdown of the Союз Советских Социалистических Республик (USSR). I saw him just after the news came out and he was quite worried what he would find when he returned home to Russia. Unfortunately, we fell out of touch, and I never found out his reaction to The New Russia. If you are reading this Fedya, please get back in touch!
Like many students of Russian, I have asked, “Are почему and зачем synonyms?”. The answer is always no, but when you ask Russians to explain when to use one or the other, they seem…uncomfortable. 
The fellow who runs the excellent “learn a new language every year (http://www.yearlyglot.com)” blog recently posted a great article on this. He explains that почему infers by what means and зачем infers for what.
Here’s an illuminating example from his article:
Q. Почему вы здесь?
Why are you here? (With this sentence, read it with the emphasis on the word “here”. It’s as if почему throws the focus away from itself.)A. Ваш магазин – единственный открытый.
Yours is the only store that’s open.Q. Зачем вы здесь?
Why are you here? (Read this one with the emphasis on “why”)A. Масло купить.
To buy some butter.
He references this article at a Russian grammar site from which helped him to understand. Check it out and see how much you can understand!
By the way, Josefina over at her Russian blog wrote this excellent post on this very same subject some time ago.
