Pro C and Pro C2 are different Pro versions, Pro C2 is a newer version but it breaks PS1 games. I can't find a list of what else Pro C2 changes.
Nowadays, Pro and LME are basically the same thing and you'll just want to use whichever works for you.
There's also a psuedo-cfw called Infinity, it lets all PSPs coldboot either Pro or LME on "6.61" (it's really a weird hybrid of 6.3 and 6.61, but for all practical purposes it's 6.61). The only downside to it is that it breaks the Pause function on the PSP Go, but you can just use PSPStates instead.