How does an OLAP cube database differ from an RDBMS? Is a cube database, simply an RDBMS—such as PostgreSQL or MySQL—configured for OLAP via a star or snowflake schema? Or is a cube database different from an RDBMS, and if so how?
Cubes are very a different beast from a traditional database.  There are several different kinds of cube storage processes depending on need (OLAP, MOLAP, ROLAP, etc.) which are all done differently depending on how real time the data needs to be.  

I actually did a webcast with another Microsoft MVP a couple of weeks ago where we talked a little about the BI process and how it impacts the traditional database, which might be interesting to you.  The recordings (which are free) aren't available yet, but should be soon.