If you look at it from a pure physical standpoint...
Flexed when possible>extended
If you look at your arm, elbow, wrist, knuckles, fingers, joints, and fingertips, you will find that playing the piano is utilizing a series of levers (known as our arm) in order to push keys and make sounds. Now then, objects FURTHER away from the fulcrum take more force to move than objects CLOSER to the fulcrum. Our fingertips are that object, so it is easier to push a key when the fingertip is closer to the fulcrum (the joint). THUS it's easier to push a key with a curved finger rather than an extended finger. And I personally don't see any benefits with playing with additional discomfort other than perhaps a different color achieved with flat fingers.