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An anti air missile has often very small wings but out performs a fighter aircraft, e.g. much higher g-load, since g-load comes from lift which is the area of wing.

Why not designing an aircraft like a missile? E.g. low drag, high agility.

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    $\begingroup$ Aren't fighters already capable of more Gs than the pilots can safely manage? Also being able to land is usually considered a plus for aircraft... $\endgroup$ – Ville Niemi Jun 27 '15 at 14:48
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    $\begingroup$ Ability to land and take off, cool, I forgot that. $\endgroup$ – user2174870 Jun 27 '15 at 14:56
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    $\begingroup$ And there is no wet meat ball inside it. $\endgroup$ – Simon Jun 27 '15 at 15:04
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    $\begingroup$ en.wikipedia.org/wiki/Messerschmitt_Me_163_Komet : rocket-powered WW2 fighter. Fastest plane in the air at the time .. for seven minutes, at which point it would run out of fuel. Also horrifyingly dangerous for many reasons such as no landing wheels and corrosive/explosive fuel. $\endgroup$ – pjc50 Jun 27 '15 at 22:25
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    $\begingroup$ @VilleNiemi Generally, no. My super hornet is structurally limited to 7.5, however, I am capable of more. More than that, once you start loading ordinance on the aircraft the max G loading further decreases. $\endgroup$ – Rhino Driver Jun 28 '15 at 3:03
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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.

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    $\begingroup$ 30 second flight time, but only a few seconds' burn? Does that mean the missile is just unguided ballistic for 20+ seconds? If so, wouldn't it be extremely unlikely to strike or even get near its target? Or are the vanes still operating, just not thrust vectoring? $\endgroup$ – ravron Jun 28 '15 at 23:55
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    $\begingroup$ @RileyAvron: It's a guided "glider" for the remainder 20+ seconds of its flight. Remember, electronics and servos don't use rocket fuel - they use batteries. $\endgroup$ – slebetman Jun 29 '15 at 5:57
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    $\begingroup$ @slebetman But, in that flight regime, it's not "[using] its thrust just as much as the lift from its vanes to change direction." So I think some clarification is needed. $\endgroup$ – David Richerby Jun 29 '15 at 8:54
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    $\begingroup$ @DavidRicherby: For clarification I use the Vympel R-73: It can engage targets which fly at the side or even slightly to the rear of the fighter launching it. While the rocket motor runs, it has to massively change its impulse to engage the target. Once that is done, it coasts and needs comparatively small corrections, for which the canard control surfaces are used. BTW, they are not driven by batteries but by a gas generator (better for long-term storage). $\endgroup$ – Peter Kämpf Jun 30 '15 at 3:53
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Most missiles do not have better agility than planes. The maneuverability of an flight vehicle is defined typically by what is called a "maneuverability graph". Aircraft generally have much better maneuverability than traditional missiles do, but there are some newer missiles that use a technology called "Thrust-Vector Control" which can potentially make the missiles more agile in some, but not all, parts of the envelope. In general missiles achieve their lethality by greater speed, not greater maneuverability.

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