What is the difference between ALTER INDEX and DBCC DBREINDEX?
The DBCC command is a consistency checker utility inside of SQL Server whereas ALTER is a DDL SQL command.  The DBCC REINDEX command was deprecated in a previous version so it is not as robust as the new ALTER INDEX command is today.  There is more functionality in the ALTER INDEX than the DBCC REINDEX commands:

http://msdn.microsoft.com/en-us/library/ms188388.aspx

Happy indexing!