In my view, it's unfortunate that these have been displacing the traditional plastic backed diapers. The whole idea of the cloth-like outer covering breathing better seems such a myth.
First, they still have a layer of plastic behind the papery cloth-like outer covering. This makes sense though, as there would have to be something to stop pee from escaping. So I think the purpose of the cloth-like covering is to reduce noise and to allow a thinner sheet of plastic to be used, where the outer covering helps to protect it from being torn.
Otherwise, I don't see many functional advantages, but several drawbacks. One is that the outer covering tends to wick moisture where it's not wanted, even when the diaper has more capacity to use. The velcro-like fasteners tend to get full of fiber strands from the outer covering, and eventually don't hold as well. So, they're not really as refastenable as claimed. And some of these brands still use tapes, which are a poorer choice because they aren't that refastenable, even with the split taping system widely used among the European designs. A well designed plastic-backed diaper with quality tapes holds much better, and can be refastened many times.
I've tried the Attends Extended Wear and the Abena AirPlus, both of which had at least one of the aforementioned issues, and didn't breathe any better, because that inner plastic still tends to trap heat compared to regular underwear. So, I think the breatheability is more of a perception than a reality. Even in the summer time, if I'm going be in a disposable diaper, I'd rather have the real thing.
I don't even mind the sensation most of the time, as I'm used to it, and regard it as part of the fun experience of wearing diapers. The cloth-backed diapers do have their place, as they are good for stealth wearing. If I had to wear diapers for real, I'd probably have both types for flexibility but would lean toward the plastic ones most of the time. I sure hope a few good ones will always be around.