What are the additional capabilities that 3D thrust vectoring (TVC) provides over 2D thrust vectoring?
Thrust vectoring allows to control an aircraft when the airflow over its control surfaces has separated. Before thrust vectoring, the range of angles of attack in which an aircraft could be controlled was rather restricted.
With the X-31, it was for the first time possible to control a completely stalled aircraft, which enabled much quicker maneuvering in order to get the nose pointed at an adversary. This was powerfully demonstrated by mock air combats with an F-18. The X-31 used a rather simple arrangement with three paddles at the engine exhaust, and this allowed to create both pitch and yaw moments. This makes it what you call a 3D thrust vectoring aircraft.
Modern thrust vectoring nozzles are round and allow to deflect the thrust by up to 20° in both pitch and yaw, making them also what you call 3D. However, this geometry will scatter radar waves in all directions, which makes it unsuitable for stealth aircraft. Therefore, stealthy designs can use only two straight vanes above and below the jet exhaust, which will create only pitch moments. This is what you call 2D thrust vectoring. When two engines are installed, differential deflection will allow to create limited roll moments, but the powerful side forces which allowed the X-31 to rotate its nose (and radar) quickly into the path of its adversary will no longer be possible.
Stealthy thrust vectoring will still allow to control the aircraft at all angles of attack, but the yaw rates possible with a regular thrust vectoring nozzle are no longer achievable. This, and the structural mass of a rectangular exhaust, are the distinct disadvantages of stealthy nozzle designs.
In short, yes; the ability to vector thrust to the sides of the aircraft, inducing yaw, does provide an advantage in maneuvering over pitch-only vectoring.
Consider the Su-30 "SM" and "MK" variants, which have 3D thrust vectoring. Here's a Su-30 at an airshow (I'm unsure of the exact variant) performing a range of post-stall aerobatics:
Many of these maneuvers require the pilot to be able to kick his rear end to one side or the other without the aid of aerodynamic control surfaces (such as the post-stall Immelmann, the J-turn, the Super Cobra which basically stands the jet on its tail in mid-air, etc) and thus could not be performed, at least not to the same degree, by an aircraft like the F-22. The utility of such maneuvers in aerial combat is unknown and probably low (standing still in mid-air is more likely to get you shot down than to provide any tactical advantage), which is one reason why full 3D vectoring hasn't been added to U.S. fighters (there was some talk about applying the systems seen on the F-15 ACTIVE and F-16 VISTA to combat planes, but nothing serious ever came of it, and most of those airframes are now slated for retirement).
An aircraft always needs "3D manoeuverability" (pitch, yaw, roll). Whether you use TVC to perform one, two or all three functions, exclusively, or in combination with aerodynamic control surfaces is a matter of design choices and optimization.
The choice truly depends on what are the requirements of the aircraft in question. Note that TVC in itself is not necessarily "better" than conventional control surfaces on the wing and/or tailplanes.
- Aerodynamic control surfaces are very efficient (with enough speed), but take some space and induce loads in the airframe. Ailerons on a wing take space that could be used for flaps. Tailplanes are a pure cost of weight and drag that you pay for maneuverability (and stability).
- TVC costs a lot of weight and complexity, but in turn has little to no direct penalty on drag. However, note that it has a rather limited efficiency (which is directly tied to the thrust of the aircraft -- which is not so good: ideally you want thrust and attitude control to be independent).
Maneuverability always has a cost. So the question really is: For a particular aircraft configuration, and for a particular requirement of maneuverability (in pitch, yaw, roll), and for a particular combination of optimization criteria (minimize weight, drag, costs, complexity,...): What is the combination of aerodynamic surfaces and TVC (and other means) that meets the requirement while minimizing the criteria?