A few things to keep in mind when deciding whether you want a unique index or constraint.
1) The index creation options other than FILLFACTOR that are available for a unique index are not available for a unique constraint.
2) A unique key can be referenced by a foreign key constraint but a column with a unique index cannot be referenced by a foreign key constraint.
3) A less subtle difference is related to the timing of validation. Constraints are checked before indexes and this can lead to a large multi-row insert/select or update to fail before modification. Often times indexes are validated at the end of a large modification. As a result it will take longer for a failure to occur with a rollback at the end of the modification.