Hello,
Root is a godlike creature that lives on the server. He can do anything and everything and is required for 99% of the software installs. Without root, you can't change program settings or things like that. DirectAdmin is a portal to the things that root can do. For example, creating a virtualhost in the apache settings requires root access. DirectAdmin has root access so it can do it.
Admin is just any other "joe schmoe" user on the server. If you log into the server with ssh as admin, you wont' be able to do much. However, DirectAdmin grants the administrators on the server more root goodies to use to control the server *through* DA. (restart services, create users etc..). The thing to remember that the admin user himself (at a system level) has the same limited access rights as any other user on the server who gets an account, but DirectAdmin just acts as a middle man to the things that root will do.
John