That would be a downgrade. Microsoft added a few (mostly Office related) enhancements to Windows Mobile and rebranded it WEHH. Windows Mobile is no longer supported by Microsoft.
Some Microsoft lifecyle links:
Thank you send me OS for upgrade MC65.
Here's the support page for MC65.
You will need a valid support contract or the device must be under warranty to entitle you to download software updates.
Operative System images are available on our support website and you need to have a valid maintenance contract to download them:
Here's the link to the WM6.5 system image:
as Dan wrote, Windows Embedded HandHeld v.6.5.3 is just a product rebranding by Microsoft and is exactly the same OS than Windows Mobile 6.5.3 with the sole difference that Microsoft support for Windows Mobile v6.5.3 ended in January 2013 while the extended support for Windows Embedded HandHeld v6.5.3 will still be available till 2020.