What's The Difference Between A Research Institute And A University?

Quite a lot, really.

A university is accredited to award degrees, at both "first" (undergraduate) and post-grad level (Masters, MPhils and PhDs).  A Uni normally has a  duty of pastoral care for its students.  Many if not most of the people doing research at a University are doing so as part of their degree course; they are not paid for it (although they may benefit from a grant or scholarship).  Universities are typically large and educate thousands of people, most of them under-grads.  

None of those things apply to Research Institutes.

In terms of the quality of research produced, A University normally benefits from the links established between departments with different specialties-- it's easier (in theory, at least) to contact someone with exactly the expertise you need, still within your own working environment.  Moreover, Unis benefit from economy of scale; research at a Uni should be cheaper to undertake because so many resources are shared.

In practice, a "Research Institute" should benefit from having a narrower focus, and higher concentration of people with the same area of expertise (so better quality research produced in that narrow field).