From what I have read, the Fairey-Youngman would be superior when used for take offs due to lower drag per unit of lift, and they could act as flaperons (augmenting the standard ailerons) at lower (WWII) speeds, thus increasing roll rates at lower speeds and control during take-offs and landings. They could also be setup for use as dive brakes. So I think they get the nod in terms of versatility and low speed handling.
But, the Fowler were simpler and lighter in weight per unit lift generated, and the installation incurred less drag when stowed vs the Fairey-Youngman type - at least in the WWII applications.
In effect the Fairey-Youngman flaps acted as Junkers/multi-airfoil flaps in one mode, and as slotted Fowler flaps in another mode. I think that in actual use both types had about the same maximum practical lift coefficients.
I do not know what the trade-offs would be for modern versions, though in so far as there are no examples that I am aware of where the Fairey-Youngman flaps are used in larger aircraft, I have to assume that the Fowler type is superior in practical applications. I believe most of the large commercial jet airliners use one form or another of the Fowler flap. The only aircraft that I am aware of that still use Fairey-Youngman or Junkers flaps are a few small civil aircraft and some ultralights.