Why is a special certification needed to land at Paro Airport?
The approach to Paro is unbelievably complicated. It is flown visually through a valley with high mountains all around and no sight of the airport until the last moment. Here is a cut of a map with the approach paths:



Approach to runway 15, marked in red, is worse in this regard. The ridge is rather close to the airport, so the final turn usually ends over the runway threshold. The wings get level with 30–50 ft (10–15 m) remaining only (here's a video). The approach to runway 33, marked in blue, allows a bit more of a straight final, but not by much. In either case the aircraft must emerge from the turn with correct altitude, speed and rate of descent. So the pilot has to know landmarks in the valley and how to judge correct approach profile by them. The approaches also often involve turning around higher in the valleys which again must be done by landmarks at places where it is known to be possible. It is this knowledge pilots have to demonstrate before they can attempt the approach (as pilots in command).

There are other airports where pilots have to receive special training to be permitted to fly there, for example the Samedan Airport near St. Moritz in Switzerland, which is similar mix of high altitude, high mountains around and visual approach, though it's still a lot easier as the airport is visible from much greater distance.

There is however nothing similarly difficult on Princess Juliana Airport. It appears approaches to Princess Juliana are only flown to runway 10 and these are over flat sea and flat beach. The runway is visible from great distance and the approach is straight. The featureless water makes it slightly harder to judge height than solid ground, but that's what the approach lights (VASI/PAPI) and distance-altitude table are for (there is no ILS at TNCM, only VOR-DME). And there are many airports with approach over water, so it's nothing unusual for the pilots. If there was approach to runway 28, it would have to be off-axis to avoid the hills on the island, but it does not appear to be used at all.



The pilot, Marek Tůma, who ferried in and for first 3 months commanded the ATR-42-500 A5-RGH, wrote a 4-part report (2, 3, 4; unfortunately in Czech) about flying in Bhutan.

According to that the training involved:


Couple of hours on simulator (in Bangkok).
Then they were permitted to ferry the plane to Paro.
Flight as observers with the training captain who showed them all the relevant landmarks in reality.
Some practice circuits.


They couldn't fly with the training captain in the other seat, because they were ATR-42 pilots and the training captain flies A320-family (and possibly still has rating for the older BAe146, but not ATR-42).

He also wrote that while on early flights when temperatures are still low the performance is good, later in the day it becomes marginal and they often needed to follow the valley quite far before they managed to climb above the mountains. Careful planning and weight and balance calculation was always necessary.