Say you have a Many-Many table between Artists and Fans. When it comes to designing the table, do you design the table like such:
ArtistFans
ArtistFanID
Even if you create an identity column, it doesn't have to be the primary key.
ArtistFans
ArtistFanId
ArtistId (PK)
UserId (PK)
Identity columns can be useful to relate this relation to other relations. For example, if there was a creator table which specified the person who created the artist-user relation, it could have a foreign key on ArtistFanId, instead of the composite ArtistId+UserId primary key.
Also, identity columns are required (or greatly improve the operation of) certain ORM packages.