h2 DB is a standalone database, that a single Java application loads directly and operates on. Usually, a single application can use a DB at a time. Other DBs of this sort are the key-value LevelDB and Java's DB4O.
MySQL, on the other hand, is designed to be more powerful, in that it is run in a server process (written in C++). Applications (a large number simultaneously) can connect to a given MySQL server (with authentication), even from different hosts.