What is the difference between "institute" and "institution"?
Nothing definitive I can think of. Institutes tend to be non-profit organizations of the liberal arts or sciences.
'Institute' is the method of referring to a firm/organisation engaged in some activity, esp., commercial or social.

'Institution' is the method of describing a widely prevalent practice, esp. marriage, etc. You must have heardpeople whine, "The institution of corruption has assumed alarming proportions".