I personally love recurves. Some of their benefits have been stated already, some have been ignored.
I do agree that there is an aesthetic value to a recurved blade. I find it can make a knife look more mean and more elegant at the same time.
I can understand how someone like The Chef has no love for them, since the recurve feature has no value in kitchen work or food prep, but for hunting, bushcraft, and general purpose the advantages are legion:
Yes, it gives you slightly more cutting surface as a function of blade length.
Yes, it will assist in cutting rope or other fibrous stands. Partly because the shape draws the blade into the material (as previously mentioned), but also because you're less likely to drop the end of your blade off your material when sawing motion is necessary.
Yes, it assists significantly in chopping (wood, not vegetables) by moving more blade weight toward the tip and by optimizing the edge's angle of attack into the material.
It also assists in draw cuts when attacking material that isn't lying on a surface. Picture slicing through a stretched hide, hanging plastic, or a T-shirt someone (who you DON'T want to cut!) is still wearing. If you stab the blade through the material and simply pull, the material will settle nicely into the concave area, and your blade is far less likely to pull away.
It can (theoretically) aid in batoning. You can keep your knife perfectly horizontal while beating on the spine and the recurve shape will still force the cutting edge into your wood at an angle.
I am *very* inexperienced with sharpening knives, but have no problems whatsoever keeping my recurves hella-sharp whether on a belt sander, a ceramic stick, a steel, or a variety of pocket/travel sharpeners. I can see how they would be annoying if you're accustomed to a flat stone (the only menthod I don't use), but my understanding is you simply use the corner of the stone and take a few more strokes. Totally worth it to me for all the perks!