The unique piece is not where the difference lies. The index and key are not the same thing, and are not comparable.
A key is a data column, or several columns, that are forced to be unique with a constraint, either primary key or explicitly defined unique constraint. Whereas an index is a structure for storing data location for faster retrieval.
From the docs:
Creates a unique index on a table or
view. A unique index is one in which
no two rows are permitted to have the
same index key value. A clustered
index on a view must be unique
You can use UNIQUE constraints to make
sure that no duplicate values are
entered in specific columns that do
not participate in a primary key.
Although both a UNIQUE constraint and
a PRIMARY KEY constraint enforce
uniqueness, use a UNIQUE constraint
instead of a PRIMARY KEY constraint
when you want to enforce the
uniqueness of a column, or combination
of columns, that is not the primary
key.