SYS owns almost everything of any importance in the database, apart from your data. As such, getting something wrong here can seriously mess up your database.
SYSTEM may own a few objects depending on the options loaded, but essentially is an administrative user. Alot of the things you can do as SYS can also be done as SYSTEM, but it does have slighly less privileges as it's not the object owner.
DBA users should be set up to do all your common DBA tasks.
A basic rule of thumb is:
Try as a DBA.
If that doesn't work try as SYSTEM.
If that doesn't work try as SYS.