It is generally accepted that articulated buses are more
suitable for shorter services with a constant flow of passengers
boarding and alighting. This is because articulated buses
promote faster dwell times at stops due to the ease of access.
As stated by others above, articulated buses have several major
advantages: they are the same height as a single-deck rigid bus
meaning that they can travel (almost) anywhere that a normal
single-deck rigid bus can and they are generally safer for
passengers.
Articulated buses also tend to share a high proportion of parts
with the corresponding single-deck rigid bus, so the extra
servicing complexity is minimal.
Double deckers are suitable for longer limited stop and express
services where the frequency of stops (and therefore movement of
passengers on and off the bus) is less. A crowded double
decker bus is likely to take much longer to unload than a crowded
articulated bus, and passengers need to negotiate a staircase in
order to disembark from the upper deck. Additionally, double
decker buses are much taller and much consideration needs to be
taken for the route that they are used on. Double decker
buses place more weight per axle onto the road and are therefore
more damaging to the road network, but on the other hand, they take
up less depot space and road space.
I hear that the bendy is more stable due to lower CG or
something to that effect.