An Agency is an administrative branch of government, e.g. Highways Agency, Central Intelligence Agency. An Institution or Institute is a professional or learned body, e.g. Institution of Fire Engineers, Royal Institution, Institute of Directors. The word Institute is more commonly used in the names of such organisations than Institution. Both words are protected in the UK: only organisations of good standing my use them in their names https://www.gov.uk/government/publications/incorporation-and-names/annex-a-sensitive-words-and-expressions-or-words-that-could-imply-a-connection-with-government#Institute

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An agency is an establishment where business is done to help others. For example, an employment agency helps unemployed people find jobs. An institute is a group that works to promote a particular cause. For example, there are art institutes and building institutes that promote these fields of learning. An institution is a corporate body recognized by law that performs a function. For example, we have educational institutions and research institutions. Hope this helps.