"What are the benefits of using a two stage trigger over single? "
The advantage is primarily in competitions where there is a minimum trigger weight. In NRA Highpower and CMP comps, the Service Rifle must have a 4-1/2lb trigger. With a single-stage, you've got that entire weight to pull. With a two-stage, you pull the first stage which is prob 3-1/2 to 4 lb and the second stage is just an additional 1/2 to 1 lb. It makes it seem like a 1lb trigger pull.
Now, if you aren't bound by minimum trigger weight, then that advantage is gone. I still prefer two-stage because that is what I have always used in shooting, from the M1/M14 and now the AR.
Some games (of which I am not familiar) the standard is single-stage. Why, I don't know. I'll let somebody else take that one.