After installing the driver, install the HP Print and Scan Doctor. It will detect the printer and have an update available. Install the update.
Windows 8.1 will not print via wireless to this printer without some modification. Download the latest drivers from the HP website for the 8620 series - not from the supplied CD. Then download the "HP Print and Scan Doctor" and run the program. It should render the printer functional with windows 8.1.
Kudos to irishfred
at HP OfficeJet Pro 8625 e-All-in-One Printer