I feel that Rive Gauche is slightly more timeless and fitting to a broader range of occasions. Jazz is delightful, and slightly more playful, and like Pasha's younger brother. Live Jazz is good, but I don't really take it seriously and rarely wear it.
I have sampled live jazz but only read reviews about Rive Gauche. From what I have read the reformulated Rive Gauche smells like a can of Barbasol shaving cream...fresh and crisp. Live Jazz had way more going on with it from what I remember.
I like all three but if I had to take one it would be Rive Gauche. Although almost everyone has voted for it I don't think it is much better than the others, just more versatile with great longevity. I would rather have all three and would probably wear them as much as each other. (Ok Live Jazz more in summer and maybe Jazz a bit less than the others.)