Its a difference in connotation, which is highly subjective and may or may not exist at all.
To me, “Not to mention” implies a bonus thing to consider that doesn’t need to be mentioned to form a complete thought; it’s just additional context.
On the other hand, “Not to leave out” is a crucial detail that needs to be there for everything that was said prior to make sense.
Both sayings are still mostly interchangeable, but connotation is the reason that so many synonyms exist.
To me, “Not to mention” implies a bonus thing to consider that doesn’t need to be mentioned to form a complete thought; it’s just additional context.
I also feel like it not only adds, but enhances or expands the initial thought. EG, “I can’t go to the store with you because I have work. Not mention that you’re in a completely different city.”
The initial thought, “I can’t go to the store with you because I have work,” is a complete thought. But the addition of “you’re in a completely different city,” enhances it. Even if that initial thing wasn’t an issue (if I didn’t have work) the thought would still apply (I can’t go to the store) because of this even greater reason (you’re in a different city).
Not to mention is an addition to a list that’s so obvious it’s an afterthought. I’ve never heard the other expression used.
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So long as the first one is said sarcasticaly.
