Yes, you can continue to use your Google Apps (once also called standard) if you still have access to it. Standard is no longer available for new accounts.
Google Apps for Work has more features so it is the next version rather than a rebrand.
They release new features all the time. See Here:
- Much more storage space, 30GB today
- Device management (mobile and Chromebooks included)
- Security and Admin control tools
- Real 24/7 phone and email support
We still use this version but have no support other than reading support forums and no control over mobile devices.