So my question is, what is the difference between statement of research interest for graduate schools admission and that for summer school?
The way you phrase the question suggests that you see this statement of research interest as a form to fill in some details. But it's not like there's a "P27.B: Statement Of Research Interest for Summer Schools" form that's slightly different from the usual "P27.A: Statement Of Research Interest for Graduate Schools" form in Annex B. These things don't exist.
I think you're approaching the question all wrong.
A research statement is a letter you write to a person with a specific purpose. So a better question would be "what is the purpose of writing a research letter for a summer school?".
The answer is pretty much the same, but instead of trying to demonstrate why you and your background are a good fit for a grad school, now you're trying to demonstrate why you and your background are a good fit for a summer school and that you will benefit greatly from that school.
There's lots of tips on the Web on how to write a good research statement. Maybe some personal pointers ...
First and foremost you need to appreciate that it will be read by a human being, not a machine. So don't make it boring or clichéd, esp. if that human being will have to read hundreds of statements like yours. Keep it concise and interesting. Many students are too concerned with filling "the requirements" than communicating; the key part of such a letter is communication, not topic lists.
Second you need to figure out what that human being is looking for from you. Such statements are not just paperwork; they have a purpose. Why are they asking you for the document? Why will they spend valuable time to read it? What is important to them?
Third you need to frame your personal context into what they are looking for.
- If you can, try to figure out specifically who is going to read it ... what kind of technical expertise they have, what are their research interests, etc. That person will probably be a postdoc or junior professor in the area.
- In the research statement, they'll want to see how the student will benefit from the school ... why that student is worth the chair space. They will simply be looking for enthusiastic students with a convincing story as to why they will benefit from the summer school. If the school is very narrow and technical, there may also be a check to make sure that the student has sufficient background knowledge to follow the topics. If you sound too expert in a topic, that's not good either; they'll want you to learn!
- In the text, discuss your research interests and goals. Relate your goals to the types of topics covered by the school. Add details; perhaps a specific lecture or lecturer or topic you are especially interested in and why. Be enthusiastic but not fake. Don't get stuck in technical details. Try to make the letter sound personal, almost conversational, like a person wrote it ... versus someone taking a template and filling in topics. Edit and remove irrelevant statements. Get feedback from peers or an advisor.