When I researched the same question, I arrived at the conclusion that the distinction was that CCP-C included flight physiology, while FP-C focused on flight physiology, both as they relate to critical care transport. Also, CCP-C(at the time) required a training course, where FP-C you could just pay to take the test.
So, a ground based job would prefer CCP-C, a flight job would prefer FP-C, and chances are pretty good that you’re going to have to get the other one at some point. This was what I uncovered at the time, which was a few years ago and I’d be a bold faced liar if I said I really deep dove the particulars. I had planned to do CCP-C(CCEMT-P course from UMBC) then take the FP-C later after studying what’s different at altitude. Once I decided on a plan that looked good for me, I moved on. There are probably many other details to consider.