Under and below both mean at a lower level and can often be used interchangeably:
The shark swam just under/below the surface of the water.
Enrique’s apartment is two floors under/below ours.
In general, under is more specific and tends to indicate a direct vertical relationship of objects; below indicates merely being on a lower level, and the objects may not be vertically lined up.
In some circumstances, only under (not below) applies, to include the idea of touching, or of being covered:
On a cold night, it’s cozy to be in bed under the covers.
The roads were all under water because of the heavy rain.
On the other hand, below, not under, refers to a place or position and means “lower than”:
Hitting below the belt is considered unfair. (Hitting to the area of the body that is lower to the ground than the belt is)
This is different from “under the belt.” If you say:
The skin under his belt was raw from abrasion,
you are referring to the skin on his body, directly touching the belt line.
Here is another contrast:
Johnny was surprised to learn that below the equator, centrifugal force goes counter-clockwise. (that is, any place south of the equator)
Under the earth’s surface at approximately the latitude of the equator lies a huge, unexploited reservoir of oil. (The reservoir of oil is in contact with the equator—even though the equator is an imaginary line, one sees it as lying on the surface of the earth.)
Below is the opposite of above. Under is the opposite of over.