I think you can easily spare yourself the trouble of remembering when to use "non" instead of "pas" for "not", since 95% of the time "pas" is not replaceable with "non". And conversely, cases when you must use "non" for not and couldn't use "pas" are very, very rare.
Actually, the only ones I can think of are "Non seulement" and "Non loin de" starting a sentence.
Non seulement il était tard, mais il faisait froid. (correct)
Pas seulement il était tard, mais il faisait froid. (incorrect)
.
Non loin de là vivait un homme... (correct)
Pas loin de là vivait un homme... (understandable but sounds like a mix of informal and formal speech, you don't really say that)
Of course you've got "non"-prefixed adjectives or nouns like "non-violent", "non-prolifération" but these are the same in English so they are basically no-brainers - which by the way you'll have trouble translating either with "non" or "pas" ;)