Essentially a sub-prime loan is an option for people that don't have prime (optimal) credit, with various types of loans that are offered to those with sub-prime credit.  These are: loans for "tarnished" credit/don't meet the requirements for a regular mortgage, as well as loans with seriously damaged credit for people recently involved in bankruptcy, collection accounts, or people that have judgments/liens against them.

A predatory loan is used to take advantage of people by charging an excessively high interest rate or too many fees.  

Bankapedia’s Take

The difference between the two is that the sub-prime is actually geared toward helping people that have a damaged or negative credit history, while the predatory is out to take advantage of people by trying to squeeze out extra money from them.  The bottom line is that the attitudes of the loan officer and lender make the ultimate decision as to whether or not the loan is good or bad.