They can often be used interchangeably. Sometimers the difference seems to be collocation- I say 'under the bed' because it seems to me that everyone else does. I use 'under' with verbs of movement- I would say that boat is going under a bridge, not below it.
Both words can mean 'in a lower position than'. But if you're talking about something being covered by something, use 'under'. So, 'I hid the key under a rock'. Or, 'officials said there was nothing under President Bush's jacket'.
Use 'below' when you're talking about something that's not physically immediately under, or not necessarily immediately under. So you say, 'below the surface of the water'. That might be anywhere below the surface of the water, not necessarily just touching it. Or, 'twenty miles below the earth's surface', definitely not immediately under it. And, by extension, we say things like, 'below the poverty line'.
In general, 'under' indicates a direct vertical relationship of objects; 'below' indicates merely being on a lower level.