Why does a missile have small wings but better agility than a plane?
First of all, a missile is operating at a much higher dynamic pressure than the airplane. After all, it has to catch up with it to do its job. Since dynamic pressure scales with the square of airspeed, the wings of a missile twice as fast as the target would need only a quarter of the wing area to produce the same forces.

Secondly, the missile uses its thrust just as much as the lift from its vanes to change direction in the initial flight phase when the rocket motor runs and the biggest course changes are needed. With an airplane, this is normally not possible, since most planes need to fly in a small range of angles of attack and sideslips to operate properly. Exceeding those will produce substantially more drag and might lead to loss of control.

Lastly, the missile has to carry only the mass for a few seconds of flight. The burning time of a rocket motor is just a few seconds for short-range missiles, and their typical flight time is about 30 seconds. Since they do not need to carry the fuel to return to base, their mass can be held down, which improves agility.