Unique Indexes and Primary Keys have similar properties. They can both be used to enforce foreign keys, both be used as bases for computed columns they can both be used as clustered and non clustered indexes and they can both be declared on multiple columns.
However they are not the same thing and should not be treated or even thought of as such. There are numerous differences. Primary Key Columns cannot contain nulls whereas the columns that make up a unique index can*. You are only allowed a single primary key but as many unique indexes as you need. A unique index is thus an alternate key.
* SQL Server allows a single null in a unique index. Oracle however allows several as nulls by thier very nature are unknown its up for debate if the Oracle method breaks unique rules or not.