Under and below both mean lower than.
If the lower thing is covered/hidden, we prefer under: (1) The suitcase is under the bed.
If lower than = fewer/less than, we prefer under: (2) There were under 200 people in the audience.
If one thing is not directly lower than another, we prefer below: (3) The sun sank below the horizon.
With measurement perceived as vertical, we prefer below: (4) The temperature remained below zero for three months.
Underneath is virtually the same as under, but only for physical uses. We could use underneath in #1, but not in the other examples.
Thanks to Michael Swan, whose Practical English Usage (OUP) cleared this up for me some years ago.
