but if they're between two people how is it different to a private phone call?
Very different, both technically and legally. Basically, cellphone is still a phone, although it uses radio, because, well... a: to listen to any specific cell phone you have to follow the owner, b: It's encrypted and c: Cell phone lobby said so.
But CB is stiill a shortwave radio, which means less bandwidth, more range. No one could or would guarantee you that you aren't heard by your mother-in-law in the next state just because she accidentaly dialed the same wavelength and ionosphere was right, let alone by FCC. Encryption is impractical with limited bandwidth (lots of technical reasons), so it isn't allowed. (and it's really a good thing in this case, belive it or not.) So, you just have to be discreet and behave yourself when you talk on CB, you are in public.