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).