These three words appear to be synonymous, but many contexts call for one in preference to another.
Beneath is somewhat more literary in use.
Under in its physical sense is rather more literal than the other two:
under the bridge means directly underneath it, whereas
beneath the bridge suggests a wider area, and
below the bridge can also mean downstream from it. It is customary to say
below par, below the belt, to go below (in a ship),
the information below, the temperature is below 20 degrees; conversely
a man under 40, under one's breath, incomes under £10,000, under the sun, under the circumstances, under one's thumb, under sentence of death, and
beneath contempt, be beneath one.