I'd say "let alone" is [usually] used in a negative sense -- for example to reinforce a statement about somebody's inability to do something by implying that the person would be even worse at doing something else: He can't ride a bike, let alone drive a car. Since it normally takes less skill to ride a bike than to drive a car, the speaker is assuming that the person involved would definitely be a bad driver. As for "not to mention", it can be used in a positive sense: The concert was simply amazing—not to mention all the fun we had chatting with the band backstage.
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