If you look at both the UMBC CCEMT-P and the BCCTPC FP-C resources on there websites, you will see in theory, that the UMBC is a foundation training course for people entering the critical care field whilst the FP-C is a difficult test designed for seasoned providers who have been working in critical care for several years.
Do not quote me on this, but I think the CCP-C draws from the same test bank as the FP-C from my understanding. The only difference being the CCP-C excludes flight ops/physiology. I believe it gives you more of the other questions to make up for that section being excluded. My friend (WT remember AJ?) took both the CCP-C and FP-C, failed the CCP-C but passed the FP-C lol. Go figure...
However, with no real national standards for this stuff and no rock solid rules, the same "Tom, ****, and Hairy's, that WT mentioned can take both of these things right out of paramedic school and ignore the recommendations, because that is all they are... recommendations. I also agree with AKF that from what I have seen, many reputable HEMS places recommended both the CCEMT-P and FP-C certifications. Even CAMTS using words like "recommends" instead of "required". There is also currently no national standard requirements for CME hours in critical care transport, only lots of "recommendations".
I just started getting into all this CCT stuff, so take what I say with a grain of salt, its just my understanding of things so far. I am taking my CCEMT-P at UMBC in a couple weeks. I am sure the guys here with CCT experience can be more helpful, I am just giving my 2 cents.