In 2015, we went to the Moulin Rouge, and in 2016, we did the Lido. Both are very similar but if you need to see one, I definitely recommend the Moulin Rouge.
First, the venue is much more remarkable, the seatings and setup are very similar, but if you want an old cabaret style, then Moulin Rouge is the one. The Lido is a bit more modern: it plays a lot with projections and lights, it has a lot of different moving stages (water fountain, ice rink, etc.).
Second, I like the Moulin Rouge better because I felt like the girls were better dancers. After the Moulin Rouge show, we definitely got a “wow, this is better than what we expected” feeling, but after the Lido, it felt more like we just saw a fashion show. I think the choreography weren’t great and the scenes were short.
As for food, don’t set your expectations too high for both places, it’s a dance show, not a gourmet thing :) I unfortunately have no photo from the Moulin Rouge show.