In the Soviet Union "Институт" was a single-profile higher education institution. So, for example, if they had all the stuff like math, physics, law, philology, economics etc. they were called "a university" but if only, say, medicine, they were "an institute".
In the first years of modern Russia many former "institutes" chose "to upgrade" themselves and became "universities" (or "academies"), sometimes really getting many new profiles, sometimes not. So there are not too many of them left now. Therefore the term "Институт" (except if not related to a higher education at all) is used in two different meanings: (a) a real single-profile higher education institution as in Soviet times, which is mostly dead now, and (b) any modern university which was "институт" about 20-30 years ago.