PICxxFxx stands for flash program memory. That is, you can electrically write/erase this memory more than 1 time. This is very useful for the prototyping stages of your design - you may have missed a bug, that you can only catch when testing the device, so if you are using an F device, you simply reprogram it and continue testing. Furthermore, you can use the same chip in the future for a completely different purpose.
The C series use the so called OTP(One-Time Programmable) memory - once programmed, the memory could not be erased. These chips are generally cheaper than their F equivalents, but i think their only purpose is for mass production. For example, if prototype the design (and test it) using PICxxFxx, a cheaper way for mass production is to use the PICxxCxx equivalent chip.
Otherwise there is no difference between the chips. You may even notice that in many cases the same datasheet refers to both the F and the C series.
Cheers