What is the difference between a research statement and a research interest?
This would depend on who is asking, of course. But I'd expect, generally, that a research statement would be more detailed with a plan for how you would go about things like obtaining funding and incorporating students, etc. A research interest might be as little as you area of special expertise, though it might be a mistake to make it too narrow. For example, in mathematics, Algebra might be a bit too wide, but not so narrow that only a few people in the world are interested. 

But, it can vary. Ask the asker for guidance if you can.