Depending on the statement, statistics and DB server it may make no difference – the same optimised query plan may be produced.
There are basically 3 ways that DBs join tables under the hood:
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Nested loop – for one table much bigger than the second. Every row in the smaller table is checked for every row in the larger.
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Merge – for two tables in the same sort order. Both are run through in order and matched up where they correspond.
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Hash – everything else. Temporary tables are used to build up the matches.
By using exists you may effectively force the query plan to do a nested loop. This may be the quickest way, but really you want the query planner to decide.
I would say that you need to write both SQL statements and compare the query plans. You may find that they change quite a bit depending on what data you have.
For instance if [Institutions] and [Results] are similar sizes and both are clustered on InstitutionID a merge join would be quickest. If [Results] is much bigger than [Institutions] a nested loop may be quicker.