A 3-piece takedown is great when it comes to portability and adaptability. You can take it down and stuff it under your seat when you need extra room in the vehicle. You can get extra limbs in different draw weights, sometimes different lengths, for hunting and target shooting. And the greater mass in the riser is a stability enhancer and that makes them a tad better for accuracy, like Pinelander said.

A 1-piece recurve is more visually appealing, at least to my eye. They're generally lighter to carry which makes them a pleasure to carry around the woods.

I like both, very much, and keep one of each kind on hand at all times. But if I were forced to have only one recurve, I'd definitely make it a takedown.