How do tablespaces differ in DB2 from databases in MySQL or SQL Server? In DB2, it seems there is a strong separation between what a database is and what a tablespace is. Where as in MySQL and MS SQL Server tablespace and database seemed to be used synonymously.

What are the differences, if any?
In DB2, a tablespace is the object that serves as a layer of abstraction between tables (and indexes) and disk.  It is similar to a file group in MS SQL Server. 

In DB2, a database is roughly equivalent to a SQL Server instance – a database contains multiple tablespaces, transaction logs, one or more bufferpools and other memory areas.  

A "database" in SQL server is much closer to a schema in DB2.