What do the classes do for your hero? (e.g. Ebon Knight, Cleric, Necromancer) The Hall of Heroes also says that it lets my hero change classes. I'd thought that it was related to the specialization but some heroes have the same classes but different specializations.
There's two factors at work here:
In the campaigns, only Main heroes can change classes. In this case, it refers to reaching 250 points on the tears / blood path. Non main heroes don't have blood / tears, and so are never eligible to change class.
While each hero has their own specialization, they also get an ability from their class. All Knights, for instance, get "Mark of the Heretic" which makes their attacks cause the target to take increased damage. Inquisitors, on the other hand, have "Holy Blades" which makes their troops deal a small amount of additional holy damage with each of their attacks.