It’s mainly a matter of which French expression is used: Non seulement is often used as the exact translation of not only when you have this kind of sentence: Non seulement … mais en plus … !. So it’s when you’re building a complete argument sentence by your own.

Pas seulement is used another way. It’s what you would say when someone said something right and you have some other thing to add to its argument.

Otherwise, there is not any concrete rule about non and pas: you have to figure out which one is used depending on the sentence and the context.

1. Dans tous ces cas on utilise pas, non n’est pas possible :

   • pas vraiment (not really)
   • pas toujours (not always)
   • pas souvent (not often)

2. ne … pas … seulement et non … seulement sont interchangeables :
   « Je veux non seulement réussir mais arriver le premier » ou « Je ne veux pas seulement réussir mais arriver le premier » veulent dire la même chose.

Metalinguistic negation

There is a particular use of non, which has been mentioned in other answers. Even though I’m not an expert, I think all similar cases I will mention here are cases of metalinguistic negation, hence the name of this section. These non negate (or expand on, or nuance) a previous claim, or hypothetic (or sometimes obvious) assumptions one could make.

1. The first form is “Non … mais …”.

   Votre participation est non acceptée mais encouragée.

   This could imply that the term acceptée was used previously. Someone thought it would convey bad connotations, and suggested that “encouragée” is the appropriate word using this phrasing.

2. This same non can be used with a few adverbs (chiefly seulement, but also véritablement, vraisemblablement, and maybe others), to introduce nuances. Examples:

   Cela est non seulement incongru, c’est aussi complètement stupide.
   Il a agit non véritablement par nécessité, mais plutôt par compassion.

3. It can also introduce a sentence:

   Non seulement il ne fait pas d’efforts, mais en plus il nous embête.

4. Or can appear in phrasings such as:

   La Terre, qui est plutôt sphérique et non planaire comme l’ont pensé certains.

5. It can also be found as an incise clause (the following is a quote including one of the very few non toujours you will ever find):

   Un des propres de leur race est, dans le feu de la jonction, de se jurer mutuellement, non toujours par convenance, qu’ils sont unis ad aeternum.

All these phrasings can also be formulated with a standard ne … pas négation, and the power of the negation is somehow lessened. For cases number 3 (and 5) it might be a bit difficult to find the appropriate phrasing, because just substituting non with pas is not an option:

1. Votre participation n’est pas acceptée mais encouragée.

2. Cela n’est pas seulement incongru, c’est aussi complètement stupide.

3. Ce n’est pas seulement qu‘il ne fait pas d’efforts, mais c‘est qu‘en plus il nous embête.

4. La terre, qui est plutôt sphérique et n’est pas planaire comme l’ont pensé certains.

5. … de se jurer mutuellement, ce n’est pas toujours par convenance, qu’ils …

Standard negation

Non is also an adverb which is sometimes used produce plain standard negation.
Here are few remarks, which I hope will help clarify when its use is possible (if not required).

1. There is a subtle difference between these two sentences:

   Sa décision n’est pas contestable.
   Sa décision est incontestable.

   The first exposes a subjective property (you cannot argue his decision, maybe nobody can); and the second exposes an intrinsic property (it’s impossible).

   To give another example:

   Cette personne n’est pas fréquentable.
   Cette personne est infréquentable.

   The first meaning (subjective attribution) is “you shall/can not get acquainted with this person”; and the second (intrinsic attribution) would be “nobody can”.

   To obtain the same effect with an adjective that does not have a pre-existing opposite, non is used:

   Sa décision n’est pas négociable.
   Sa décision est non négociable.

   In particular, the latter will be perceived as more absolute/authoritative.

2. Using non before an adjective which admits a straightforward opposite would sound odd, use difficile instead non facile, basse instead of non haute, etc.

   Though, there are lots specific cases which might require some more care, for example, réaliste can be negated as non réaliste even though irréaliste exists, because that word could also be interpreted as “pertinent to irreality” (which is not quite the same thing).

3. The subjective/intrinsic distinction also applies when adjectives are used as epithets:

   Une personne (qui n’est) pas fréquentable. (subjective)
   Une personne infréquentable. (intrinsic)

   Une décision (qui n’est) pas négociable.
   Une décision non négociable.

   Note: I think there is something slightly wrong with the omission of “qui n’est” in a formal usage, but you will of course find it omitted quite often.

4. Adverbs.

   Non is also used to negate adverbs. In particular it assumes that negated adverbs don’t have pre-existing opposites. For example faussement is to be preferred to non vraiment.

   Some subtle distinction between a subjective or intrinsic intention is sometimes still present.

   Un rapport non clairement établi entre fréquence et durée
   Un rapport qui n’est pas clairement établi entre fréquence et durée

   The topic of first phrasing is a correlation with an unclear footing while in the second sentence the topic is a correlation, with the additional information that it has an unclear footing. Is the difference noticeable?

   Also, the reason why “non toujours” and “non souvent” are extremely rare is that occurrences are generally not an intrinsic property of what is being described. Moreover “non souvent” would more likely be replaced by “rarement“.

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" 😉