Part 4.1 here has all the details about the movement and the watch.
The exact same movement
This is just a brief comparison of the previous and current version of the Reverso Duo in stainless steel on a bracelet. JLC's leather straps are without doubt of the finest quality but are priced at over 400 dollars retail. So the bracelet is usually a good choice at initial purchase and a fine quality after market strap works very well.
The key differences in the two models are in the dial and the pusher. The movement is the exact same in the two iterations and from what I can tell the bracelets are the same.
Old dials (courtesy of JLC via PuristSPro)
New dials (note the wave pattern on the white dial is not visible)
(Courtesy of JLC via PuristSPro)
The new white dial has a waved guilloche pattern in the center portion of the dial and a round seconds hand dial while the older version had a diamond guilloche pattern and a rectangular seconds dial. The small textured dots around the perimeter of the dial have been removed. Lettering is still printed instead of applied and personally I prefer the older dial with the finer details.
The black dial has also been simplified, the previous version had a circular wave pattern center dial with a diamond patterned outer dial. The markers were also applied. The new version has a plain center dial and printed markers. I'm not sure which one looks better but given the classical nature of the watch, I personally prefer the older version, especially since I prefer applied to printed markers. Those seeking a simpler watch, especially to be worn on a strap, will likely prefer the older dial with finer details to contrast with the strap while the bracelet may look better with the newer dial, but that's just a matter of personal opinion.
The pusher is probably the biggest source of debate amongst collectors. The old version worked just fine and even came with a tool with which to push the pusher, advancing the hour hand of black dial in one hour increments. The new version is more "user friendly" since it can be pushed like a chronograph pusher. Granted the recessed pusher could be made to be water resistant and is the method still widely used for adjusting functions such as perpetual calendars and moonphases, I neither understand or appreciate the change. It makes the watch less symmetrical and proportional, which is probably the biggest aesthetic selling point of the Reversos (remember the golden ratio of 1:1.6).
Overall, I prefer the older version a lot more, the only advantage I can really give to the mark ii is the 3 atm water resistance and 1000 hour control test, but those are largely meaningless for a watch that was never mean to be worn around water.