Errors backing up SQL Database

I do not have this issues with SQL server 2008 using the same permmisions above. Wondering if there's something different in 2012 that am not aware of. 

It has to do with security/permissions applied by SQL dba, not with SQL version.

Ask SQL dba to change NBU Client Service Account to same account that is used to start MS-SQL services.

Since the NBU client server seems to be running under the LSA (NT AUTHORITY\SYSTEM), this technote may answer your question:

http://www.symantec.com/docs/HOWTO85432

An excerpt from the aforementioned technote:

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For SQL Server 2008 and earlier, the sysadmin role is automatically applied to the NT AUTHORITY\SYSTEM and BUILTIN\Administrators groups. You can use Local System for the logon accounts for the NetBackup services.

For SQL Server 2012, you must first apply the sysadmin role manually to the NT AUTHORITY\SYSTEM or the BUILTIN\Administrators group. Then you can use Local System for the logon accounts for the NetBackup services.

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Try to run batch file for this backup from client and check if its running ?

NBU MSSQL AGENT> File> manage script files  to run that batch file.
 

if backup runs successfully then try following:

use domain accout having admin privilages to start netbackup client service and netbackup legacy network services.

Also create a role in MS SQL with sysadmin privilages and connect to instance to backup SQL objects using this user and then try to run the backup.