Answer:
PPP (Public Private Partnerships) is, as the name suggests, a broad term for partnerships between public and private bodies. These are nothing new and you can see examples of these (although never actually named thus) in 1950s inner cities in UK where construction companies entered into contracts with local authorities to build sky-rise "council flats", replacing the slums that were growing at the time.
PFI (Private Finance Initiative) was a concept conceived in UK and Australia in the late 80s and implemented in the early 90s. This saw private companies (and private funding) being used to DBFO (design, build, finance & operate) social infrastructure projects (schools, hospitals, etc). Private funds are used to design and build the assets and government money is used to pay them for the availability of the asset. This payment is also used to repay debt to the banks. At the end of the concession period, the building is turned over to public ownership.
PPP/PFI, in the UK, can be used interchangeably when referring to the PFI definition above, however PFI came under some bad press in the late 90s and the term PPP started being used more often (possibly because it had the words "public" and "partnership" in it). Elsewhere, PPP tends to be used (again, possible for the same reason), however, whether the asset is returned to public ownership at the end of the concession or not is down to the individual State.