I recently started riding in a monoflap jumping saddle, so I can tell you about my experience.
Advantages: Number one, less material between you and the horse. The saddle was bought for a Gypsy Vanner that I leased for many years, and as she was so wide, it is really nice to have one less layer. It lets you have a closer feeling with the horse (in my opinion). I think that is why lots of dressage saddles are mono flap, though I could be wrong. Also, there is less to clean. It didn’t occur to me until afterward, but that is a nice change.
Disadvantages: The biggest disadvantage is that it is harder for me to tighten my girth while riding. On the aforementioned Gypsy Vanner, I used to have to go up an average of 8 holes after mounting! I’m getting better at it, but it is harder for me to get to the girth. You also need a short dressage girth. Luckily I had one lying around from an old dressage saddle, but it is one more thing to buy!
Overall I am a fan of the monoflap saddle that I am using, but it is also the nicest saddle I have ever ridden in, so it is hard to separate the two things!