"The priests are attempting to sing."
Translation:Prestene forsøker å synge.
In certain contexts, like the example above, they are interchangeable, but prøve is an older word in Norwegian (in use in old Norse) and has heaps of meanings while forsøke is a bit younger and it's uses are narrower. Mostly you can also say prøve if you can say forsøke, but the other way around might not work.
Forsøke: try, attempt
Prøve: control, test, try, attempt, tempt, experience etc.