I don't see much difference between the two (I might be wrong).

Both unique Constraint and Unique index are enforced using an Index.
When oracle reports an unique index violation it will report that as a
constraint violation like : ORA-00001: unique constraint (SCOTT.DEPT_UI1) violated.

When you create a primary key constraint on a table, the uniqueness of the data
is enforced using an unique index. So if you try to create a duplicate
record that violates the uniqueness of the index, you will get
a constraint violation error.

Hope it helps