I think it really depends on what your research interests are and the opportunities available through REU. When I was considering both, I did an extensive search of REU sites and did not find one that directly fit my interests. Instead, I decided to seek out my own internship through a lab that was conducting research directly related to my interests. I found this to be a great experience and the networking gained through the lab and professor I worked for was invaluable in the grad school application process. Many professors commented in my interviews how impressed they were that I took the initiative to seek out the opportunity and also commented on the letter of rec that I had from the specific PI I completed my summer experience with. I would definitely suggest this option if you have the funds and support available to pursue something like this. Many of these opportunities will be volunteer positions and you will be sacrificing a summer of income. I realize that this is not a good option or possibility for everyone. It is very convenient if the lab you would like to work with is near your hometown or friends you could stay with so that you are at least not paying for housing if you are pursuing an unpaid research internship. I was coming from a small liberal arts college in a rural area that did not have a lab structure for research opportunities or access to many populations other than students. The experience I gained with my population of interest through my summer experience was especially helpful in backing up my research interests in my personal statement.
REU has its own benefits in that you can get funding for your research experience. If there is an REU site where you are particularly interested in the research and the experiences offered through REU will benefit your long term goals, I would go for it. It basically comes down to what your interests are and the opportunities available for you.
Regardless of what opportunities present themselves and where you end up, make the most of your summer research experience and show initiative, passion, and work ethic. I would say this is most important in terms of the grad school application process. Most schools will care more about what you made of your summer experience (i.e. involvement in data analysis if possible, presentations, interaction with specific populations, etc.) then where you worked. However, experience with a well-respected professor in your area of interest will further enhance your application, pending on your performance. The combination of a productive summer experience and the recommendation from a professor in your specific area of interest is very attractive in the graduate school application process.
Whether you pursue this through REU or independently is your choice and I am sure both are good options. It is not my impression that having the name REU on your CV will greatly enhance your application. I think the other factors I have mentioned should be taken into greater consideration when evaluating the two different opportunities. There are costs and benefits for both, but ultimately you will get out of the experience what you put into it wherever you end up.
