Is going to institute the same as going to grad school?
If you want to get a PhD you need a university to award the title. You can work with a research institute during your PhD, if the institute is somehow affiliated with a university. Ideally, you will get paid from the institute, work with researchers there but you still have to pass the required courses at the university, do the necessary publications. If all goes well, the PHD title will be awarded from the university.

To elaborate a little bit more. Research in institutes (I will speak from an EU perspective) is usually funded by "soft" money awarded from a grant or research program. The outcome of those projects is not only papers but also deliverables (i.e., written reports), online demos, even full fledged server applications and infrastructures. Thus, working in a research institute is not just producing papers (you do that as well) but also produce applications, write deliverables, prepare proposals and so-on. Conclusively, in research institutes, there are junior people working on the various projects for the more mundane "programming" tasks. Those people are more "disposable" (i.e., they work only during the duration of one project). Contrarily, there are people with a PhD that might work in this institute for years, have worked in multiple projects and have a more active role (they supervise the junior people, write proposals etc). Although many of them are not permanent employees of the institute (they are still paid from "soft" money, as well) they can collaborate with a research institute for many years.

Where do graduate students fit in all this scheme? If the institute is closely affiliated with a university, the PhD students of the university sometimes are assigned the junior roles in some projects, in order to get funding. As years go by, the PhD student (towards the end of his PHD) becomes a expert in his area (which might be close to the project's scope) and he may be assigned more senior roles.  

So, if you want to do research in such a institute for a long time, you will need the PhD title, sooner or later.