Use the OUTPUT Inserted.* INTO #tmp on an INSERT when you need the IDENTITY / NEW_ID / SEQUENCE values for the new records (e.g. for further processing / creating referencing entries in child tables ...).
Uuse the OUTPUT Deleted.* INTO ... to move data from one table into another one, either for archiving or because you are refactoring your data structure. Or you can use it for some logging / auditing stuff, but in this case I would prefer triggers.
PS: you can use OUTPUT on a MERGE statement too, there you can refer to Inserted.*, Deleted.* and a special "variable" $action, that will be set to INSERT, UPDATE or DELETE, depending what happend to this row in the MERGE.
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