How do the uninterrupted and interrupted flaps compare?
The two designs are indeed based on compromises.

Earlier Boeings were designed for high-Mach cruise, which means high sweep and the need for elaborate flaps when flying at approach speeds. The 'thrust gate' reduces the thrust/flap interference and drag, especially in a go-around. It is also a good choice when more powerful engines will be offered as the MTOW increases.

Since the A320, Airbus have used high-lift planform layouts, which allow the continuous single-slotted flaps to work. An A380's approach speed is slower than a 747 or a 737, despite having single-slotted flaps. Owing to its high-lift wing.

When Airbus designed the A330/A340, it helped that they launched the bigger variants first. A targeted (tailored) design is not good when the aircraft would later be stretched, which would require a re-design of the flaps, i.e., if the shorter variant comes first, then 'over-designing' the flaps and having a thrust gate is a good choice.

Interestingly for the A350-1000 (a stretch of the -900), Airbus had to increase the wing area by 4% by modifying the wing trailing edge to maintain the same approach speed (flightglobal.com).

Losing the gap does save weight, complexity, and by extension, maintenance. But the limitations are:


Slower cruise speeds (not a problem since it's by design, flaps follow wing).
Thinner spoilers due to thicker flaps, they have to be made stronger.
Must demonstrate controllability during a go-around.
When stretching a variant, a flap redesign will be needed (ala A350).
Limits the flap span to allow for the innermost aileron to be inboard enough to not cause aileron reversal (lots of wind-tunnel and CFD models help fine tune this).


Own work based on scaled drawings from boeing.com and airbus.com.

Comparing the wings of the A330-200 and 777-200(ER):


The maximum operating Mach numbers (MMO) are 0.86 (A330) and 0.89 (777). Note the higher sweep on the 777.
The MTOW and MLW of the 777-200ER are 55 and 31 tonnes heavier, further evidence to the higher wing loading of the 777, which would indeed need more elaborate flaps.
Keeping in mind in the development cycle, the 777 was stretched, while the A330 was shrunk.


In short:

Boeing tends to prefer higher MMO, higher MTOW, and to offer the base-variant then scale it up. All of which would benefit from the 'thrust gate' and the more elaborate flaps. The overall trend however shows Boeing going for simpler designs:





This answer is based on a publication by an Airbus engineer:


Reckzeh, Daniel. (2017). Aerodynamic Design of Airbus High-Lift Wings in a Multidisciplinary Environment.