I can't understand the difference between "don't forget" and "not forget". How do they differ in meaning and how should they be used in the text?

This is a negative imperative sentence, so "don't" (or "do not") is required. "Not" would simply be wrong.

Not forget is not grammatically correct here. It is just there as an option for learners as a wrong option since you have to use do or did when using not. So the grammatical rule happening here is that you need to use do or did in a negative sentence.

You need a verb. "Not forget" has no verb and, thus, doesn't complete the sentence. "Don't forget" gives you the verb you need there.

A negative imperative sentence (in current Modern English) requires a form of 'do not'. https://dictionary.cambridge.org/grammar/british-grammar/imperative-clauses-be-quiet