Nothing definitive I can think of. Institutes tend to be non-profit organizations of the liberal arts or sciences.
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Whereas the word ' Institution ' is meant to refer to the wider concept or scope for social-regional, national, global or,even, universal welfare or happiness causes or ideals ; the word ' Institute', is, me thinks, to be used for any closed set-up organisation, aiming at and or working for any social, profiteering or otherwise -- Popuri Satyanarayana Murthy, AGM(RTD.) BSNL, Government Of India