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" ;)