Many people tout thrust vectoring as automatic "I win in a dogfight" button for any aircraft equipped with it. But reality about it is far more complex than that.
In reality, thrust vectoring is only useful for supersonic and low-speed subsonic maneuvers, but for similar reason - ineffectiveness of control surfaces.
SUPERSONIC FLIGHT AND MANEUVERING
In supersonic flight, aircraft becomes stable; moreover, classic tail control surfaces loose effectiveness in that area due to interaction with wing. There are solutions, however - delta wing, canards and thrust vectoring.
With delta wing, control surfaces are at wing itself, meaning that air will reach at least part of control surfaces.
Close-coupled canards help energize the wing, allowing for better lift at high AoA; moreover, flow remains connected to wing for longer, allowing it to reach control surfaces at end of the wing (similar but weaker effect can be achieved via LERX).
Long-arm canards are positioned in front and above of the wing; thus, there is no interaction with wing, allowing them to remain effective at any speed.
Thrust vectoring helps alleviate problem of tail ineffectiveness for tailed fighter aircraft, thus improving maneuverability. Another benefit, which applies to all configurations, is allowance for better positioning of aircraft relative to airflow during normal flight, thus lowering fuel consumption.
LOW SPEED MANEUVERS
At low speeds, below 150 knots, control surfaces are not very effective due to air flow over them being insufficinetly fast. Moreover, aircraft does not have large inertia, allowing for fast changes of direction.
In high subsonic flight, however, we get following situation:
[Only admins are allowed to see this image]
At these speeds, TVC-equipped aircraft (lower one in animation) actually turns at same rate as non-TVC aircraft; however, TVC increases angle between aircraft and air flow around it (Angle of Attack, abbreviated AoA), resulting in increase in drag for no decrease in diameter of turn (that is, maneuverability), resulting in increased energy loss during maneuvers, leaving aircraft more and more vulnerable to missiles and gunfire as fight drags on. In short, aircraft does not fly in direction its nose is pointing at.
That is problematic because most combat happens at high subsonic / transonic speeds; even if combat starts at supersonic speeds, energy loss will cause aircraft to slow down to subsinic speed after not much time. At the other end of spectrum - low subsonic speeds - aircraft does not have energy required to evade missiles, leaving it vulnerable.
Another danger happens at turn onset - when TVC engages, aircraft using it rotates in air, rear end of aircraft drops, and aircraft itself sinks in air, resulting in huge energy loss, which can be exploited by skillfull pilot. Post-stall maneuvers - one of main "benefits" of thrust vectoring - are useless for exactly that reason.