What are the benefits of using thick trailing edge flaps?
The best flaps (aerodynamically, as you ask) will be complicated and elaborate. But complicated flaps add weight and maintenance cost. Airbus wanted to have a variable camber wing for the A340 (Flight), but they didn't pursue it to the end, due to the complexity.

As for the aerodynamic benefits of simple flaps, there aren't any, as the other post shows: slower MMO and smaller MTOW.

Slower MMO is because of the smaller wing sweep because of the simple flaps because of needing a good [slow] approach speed. MTOW is because simple flaps don't carry as much off the ground.

The simple flaps are thicker because:


The flaps are shorter chord-wise: the aerodynamic load (lift and drag) acts on a smaller surface area, compared to the complicated multi-slotted flaps.
Thin flaps would produce even less lift (thanks to @PeterKämpf).
The lack of a thrust gate: they need to be thicker to withstand the jet blast.


Also see comment by Peter Kämpf.

Comparing the A380 and 747-400 flaps. Note the thrust gate and the triple-slotted flaps of the 747.