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.