For example, flaperons (flaps + ailerons) and elevons (elevator + ailerons), are all individual control surfaces but play the roles of two different control surfaces. What is the point of making a single control surface, undertake the functions of two?
Flaps which only extend over part of the wing will cause lift changes mainly over that part of the wing, which will result in a lift distribution over span which is far from the elliptic ideal once the flap is deflected. Therefore, it makes sense to move the ailerons in the same direction, especially when the inboard flaps are simple camber-changing flaps.
In order to give an aircraft wing docile stall characteristics, the planform should not be elliptic, but be "fuller" (= having more local chord than the ideal ellipse) at the wingtips. With an elliptic planform and an elliptic lift distribution, the local lift coefficient will be identical over the whole wing, and gusts might easily cause a local stall at one outer wing in slow flight. The consequence will be an uncontrollable roll, which can ruin the pilot's day when occurring close to the ground. Therefore, it is better to leave some margin by increasing local chord and adding some washout towards the tips.
Now this wing will have an elliptic distribution only at one angle of attack. If the angle of attack is higher, the lift distribution will be fuller, and if it is lower than the "elliptic" angle of attack, some lift will be missing towards the tips. Both conditions will cause higher induced drag than necessary. Flaps help by adjusting local camber such that if they are set correctly, the spanwise lift distribution will be much closer to the elliptic ideal than without flaps. This requires the flap schedule to make bigger deflection changes towards the root - watch any modern high performance glider change flap settings, and you will see what I mean: The flaperons will change deflection less that the inboard flaps. This has the additional benefit of leaving more margin for aileron deflections.
This is the disadvantage of combined control surfaces: What is needed for one function is no longer available for the other function, because the maximum useable deflection range is limited.
For the nitpickers: Please understand the term "elliptic lift distribution" as "elliptic circulation distribution". Then all is well.
Several good reasons depending on the design and intent of the aircraft:
Mechanical simplicity. In your case of "flaperons", having a single set of control surfaces do two things decreases the number of moving parts the plane has. Simple is best; the simpler your aircraft, the less can go wrong (and the lighter it is, and when you're trying to get something off the ground, lightweight is good too).
Durability. Even when mechanical simplicity isn't as much of a concern and the craft has distinct flaps and ailerons (in multiple sections), making the ailerons react to flap position as well as stick/yoke input can help normalize the forces across the wing, reducing shear moments along the wing spar and other stressed sections of the wing, decreasing the potential for fatigue and failure. Some commercial airliners, particularly Boeing designs, have rear-wing control surfaces in three sections; innermost are a set of multi-tier flaps, then in the middle are "inboard ailerons" which act as ailerons at cruise speed but flaps at takeoff/landing, then the "outboard ailerons" which are used only at low speed.
Aerodynamics. Every square inch of surface area of the plane increases drag. Designs like the V-tail reduce total surface area by eliminating an entire control surface and stabilizer from the empennage, using the remaining two as "rudder-vators". This Beechcraft V35 is known as a very efficient cruiser especially at altitude (unfortunately they also have a bad rap for squirrelly roll behavior and a series of structural failures early in production, leading to their moniker of "Doctor Killers"):
Stealth. The YF-23, which lost the ATF competition to what would become the F-22 Raptor, used a similar rudder-vator configuration because it reduced the plane's radar cross-section compared to the 22's parallel elevators. The resulting plane was actually stealthier (and faster) than the Raptor, but less agile.
The B-2 Spirit, obviously, has no tail at all, so the primary control surfaces do quadruple duty as ailerons, elevators, spoilers/air brakes, and rudder (by opening the air brake on one side or the other). The craft has separate flap systems, so the primary control surfaces are only "spoilaileruddervators":
Computer control. The F-16 was designed with "elevons" instead of traditional aileron/elevator controls for a number of reasons, one of the primary ones being to reduce the required complexity of the F-16A's original analog FLCC. The only control surfaces on the wings themselves are trailing-edge flaps which are also under FLCC control and extended in any high-AOA situation. Delta-wing aircraft without canards, like the Mirage 2000, use elevons by necessity (the Rafale and Eurofighter use canard elevators but supplement them at high speeds or AOAs with the elevons).
Further speculation: Fewer control surfaces overall means fewer moving parts, less mechanical complexity, less weight, possibly less drag, and so forth.
The downside is that the control system must allow the combined control surfaces to be controlled in a way that makes sense to a pilot. This can be done mechanically, but especially in aircraft that are fly-by-wire anyway, combining the inputs in software has zero weight or mechanical-complexity cost.
Glider designers use flaperons frequently because the chord is so narrow and the wingspan so long that the ailerons have to span the length of the wing to have sufficient roll authority. This leaves no room to add flaps, so they are incorporated in to the ailerons.
This is speculation, so give it the credibility it deserves....
If the required aircraft performance dictates ailerons to be essentially the full length of the wing (as in the plane you pictured), there may not be sufficient room for flaps to also be as effective as needed. So the solution is one control surface long enough to satisfy both needs.
