Small (_low aspect ratio_) wings have the following advantages:

 - Smaller wings create less structural stress 
 - Aircraft can roll more easily with smaller wings
 - Smaller wings create less parasitic drag

However, smaller wings create less lift due to their smaller surface area.

Small wings have many use cases in modern aircraft, such as modern jet fighters, since those aircraft are built to take evasive maneuvers at high speeds. 

As you might expect, aircraft with small wings need powerful engines to generate lift. At an extreme end of this spectrum is NASA's X-15, a very small hypersonic experimental aircraft, powered by two liquid-propellant rocket engines. Making the wings on that craft smaller would essentially demote them to large stabilizing fins and the craft to a manned missile.

I hope these two illustrations have shown that there are few "totally wrong" wing sizes, and that modern aircraft designers take care that the right wing configuration is used according to the missions that the aircraft will undertake.
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