When possible, are clustered indexes that are declared as unique more performant than clustered indexes that aren't declared as unique? When applicable, will using the UNIQUE constraint for a clustered index help improve performance in any way under any scenario?
Adding to Tibor's answer, from the optimizer's perspective, key information can be very useful. This sometimes manifests in obvious ways like avoiding a Stream or Hash Aggregate operator to perform a redundant grouping operation.

Less obviously, the presence of a declared and enforced key can allow the optimizer to explore logical alternatives that are only valid when a key or functional dependency is known to the optimizer. The optimizer does not report when it could not explore a particular option due to the lack of a unique index.

From an execution performance point of view, one example of uniqueness helping performance is when a b-tree seek becomes a singleton lookup rather than a seek plus range scan:

For a unique key or constraint, the execution engine knows it will either find a row at the b-tree seek point, or it will not. When the index is not unique, the engine must scan forward or backward from the seek point to find any further matches. There is an overhead to doing that as I demonstrate in my article Enforcing Uniqueness for Performance.

In general, you are much better off enforcing a key than not. This protects data integrity as well as the other benefits mentioned above.

Related: Should I mark a composite index as unique if it contains the primary key?