I did look, but didn't really find a clear definition. I'd always assumed there was a major difference between the two.
I thought piles were often quite small, very extended parts of blood vessels that found their way out - like grapes on a vine - the nodules containing think or dried blood. It is easy to see how these might be strangled by a band. (Don't try this at home folks)
A haemorrhoid I thought, was often in the form of a large swelling outside the sphincter, but having traveled under the skin through or past the anus in some way. I think this is the type that is very painful.
Clearly, there is a blood vessel under the skin, and it is this that can be coaxed back. Doing the same with piles would be simply hiding them - very temporally.