Q. The difference between 'Don't do something' and 'Not do something'
First lets put both these sentences into BASIC English.
'Don't do something' = Do not (to) do something
'Not do something' = Not (to) do something
Do not do something in this case, I believe the FIRST "do" which is an auxiliary verb is added to emphasis the point of "not doing". However it is commonly used with another verb to form questions and negative sentences.
do auxiliary verb (FOR EMPHASIS)[ + infinitive without to ] used to give extra force to the main verb:C.E.D. It is also used with another verb to form questions and negative sentences, including negative orders.
The second "do" of "do something" is a full verb used meaning to perform, take part in, or achieve something
The verb do can be both an auxiliary and a full verb. As an auxiliary we use do in negative sentences and questions for most verbs
So we have a short, stand-a-lone, emphasised negative command. A sentence that forms a command is known as an imperative sentence.
The imperative mood is one of the four main verb moods in the English language. As opposed to verb tenses, which indicate time, moods indicate states. The imperative mood indicates the state of commanding.Grammarly
** Obviously the phrase not do something has the same meaning as in the first sentence (but without the emphasis) However, it must be used in context it does not work as a stand alone statement. It also may or may not form an imperative sentence.
Answer. Therefore the two examples are not interchangeable and the second cannot be used as a stand-a-lone sentence. Finally the mood of the phrase may or may not be the same depending on context.
I will not do the washing today, as they have forecast rain later.