Unique Index and Unique Constraint both are the same in terms of working in Tables. They achieve same goal. Performance wise also both are same.
ALTER TABLE table_name ADD CONSTRAINT constraint_name UNIQUE NONCLUSTERED(column_names) ON [PRIMARY]
CREATE UNIQUE NONCLUSTERED INDEX index_name ON table_name
(column_name) ON [PRIMARY]
In both syntax are different the effect is the same. Unique Constraint creates Unique Index to maintain the constraint to prevent duplicate keys. Unique Index or Primary Key Index are physical structure that maintain uniqueness over some combination of columns across all rows of a table. It is a convenient way to enforce a Unique Constraint for SQL Server.