From what I gather the CB model is popular among Cowboy Action Shooters. The heavier barrel swings a bit smoother, or something along those lines. Sort of like why you'd shoot trap with a solid O/U rather than a light single barrel: The light guns swing a bit TOO quick sometimes.
For me, I took the C version. It's a bit cheaper, though I can't remember by how much, it works for my plinking fun, and when I don't think I've had enough trigger time on the 870 the 1894C takes its place in the bedroom closet as the go-to long gun.