The latest available version for you to currently upgrade to without any hesitation is 6.5.6. Bug fixes, and feature enhancements as well as further OS support.
Whether you can upgrade to NetBackup 7 is down to your agreement or license schedule. Sometimes you receive a notification via email that you qualify for the upgrade.
Installing admin console does on a machine does not require a license. Unless you intend to backup that client its installed on.
Practical benefits - I refer you to the release notes which show you the benefits.
The major change betweeen 6.x and 7.0 is the drop from support of most 32 bit servers as master and media servers.
Everything to get you started is found from the main NB support page, if you are considering NetBackup 7.0 specifically the Upgrade Portal page should be read (http://www.symantec.com/docs/TECH74584)
Main NB Support landing page
http://www.symantec.com/business/support/index?page=landing&key=15143
