The main difference from being an invited, visiting student, an REU (Research Experience for Undergraduates), and a summer research program is usually related to funding issues. REU is what the National Science Foundation (NSF) calls it's program, and the funding provided usually only supports "U.S. citizens, U.S. nationals, or permanent residents of the United States". Students who are part of the program usually get a stipend as part of the package. Some programs who are advertising an REU still accept international students, it's just that the funding for those students are not provided by the NSF - read 'terms and conditions' if there is a formal program for more details.
Being a visiting student or being in a summer research program is similar, but generally supported by institutional resources (which may just be administrative, with no funding).
However, the meat of the issue is what you get to do while you are there. If you are interested in graduate school, then ideally you would work with a professor in a group on a real research project. Ideally it could potentially lead to a publication down the road, but this is neither necessary nor guaranteed - but it's good to work with that as a long-term goal to help guide and motivate the work.
I was part of an NSF REU on the east coast as well, but based on my conversations with other students the benefits I'll mention below still can apply.
So the main advantages possible are:
- working hard and doing good work that could be the grounds for a very supportive letter of recommendation, ideally from someone who is well known (and from a well known University is good as well)
- networking, which is to say expanding your social circle and developing contacts with people with similar interests and direction (very hard to do without physical presence for an extended period of time)
- building your confidence and making you more comfortable in your statements and future work
- learning more about different aspects of the research process, which you may not have seen in this particular way before
- living in a place you've never been to before, which is not only valuable as a maturing and learning experience in its own right, but can help you decide if you might really be OK with moving to such a place for multiple years away from everything and everyone you know
- if you work hard at networking and socializing (while still doing great work), you can potentially meet a future advisor, or even just someone who will help you with your application materials, statements, give you an introduction to someone else, etc.
- it can be a great way to see behind the curtain and experience both lots of good things, and even more importantly, lots of unpleasant and bad things. Academia is no utopia, and I don't think it does anyone any good to go into a graduate program thinking there aren't all kinds of messes and complexities. This could be a chance to see more of that first hand, to help you be more realistic.
- beyond just getting into a graduate program, the networking and experience can aid you in doing well in research, knowing people at conferences, etc.
While I personally strongly encourage everyone to seek out such opportunities, it is up to you to consider the costs and decide what is right for you. On the downside:
- it is often expensive. Even though I was funded quite reasonably, I would have had lower expenses and had higher income if I had stayed home and worked for the summer.
- it can be hard emotionally. You are in a new place with people you don't know, away from family, friends, food you are used to eating, the comforts of home, and more. It can be so great to meet new people and form new friendships, but that doesn't erase the hard parts.
- it is very time-limited, which means everything is compressed. It's enough time to start to get comfortable before you have to leave, while also being long-enough to develop homesickness.
- you are a short-timer, and just because you are there does not mean you are automatically an insider or in-group member. Some people will be accommodating and friendly, and other people may ignore you because they know you will be gone soon anyway. Some people will take time out to meet with you and give you advice, some won't give you the time of day.
- there is no guarantee of anything. You may find people you thought you would work with become unavailable, the project disappears (one person I know at a different place found the 'project' was unavailable, and they ended up having to make their own project by talking another faculty member into letting them work on their stuff instead).
- if you are doing things in the academic off-season (like summer in the US), many people will be gone regularly, lots of vacation and conferences, most students won't be around, many food places on campus might be closed, etc. It's just different.
I'm very glad I chose to do a summer REU, and things worked out nicely for me - I'm very happy with the results. I can't tell you the future of how things would go for you - but if you are excited (if not nervous) for the potential positives, willing to work hard and get way outside of your normal comfort zone, and are ok with the potential negatives and willing to roll with whatever difficulties you find, I think you would be a great candidate. Budget out the costs and make sure this expense won't prevent you from having the funds to apply to grad school (it's expensive, especially in the US), travel for potential campus visits if you get accepted, have your own personal living security, etc.
Whatever you choose, good luck!