The classic uses webbing material similar to(maybe the same as) a BP or other BC - perhaps a little thinner/flexible actually. It uses sliding buckled to adjust the shoulder straps. I don't think I can adjust mine without taking it off, or at least supporting the wrights so that there's slack in the straps. Pockets are movable (sliding buckle again) along the waist band. The sliding buckle adjustment means that there's no loose ends dangling, but you have to adjust them up to far, I suppose it could put the hard plastic buckes underneath your BC shoulder straps which may or may not be an issue.
I was loaned a W&T II during my OW checkout. Same problem - needed a bunch of weight for a thick suit/cold-water/fat-guy/no-butt situation. It's more of a fabric based harness, not webbing. It's shoulder straps are the "pull down" adjustable kind though, like a jacket BCD. I suppose this makes it easier to cinch them up after donning and while in the water. I haven't used the II since checkouts though so I don't know the long-term pros/cons. I suppose it's overall a bit less bulky, and the different kind of shoulder straps might make it harder to confuse with your BC straps, which I confess happens now and then when trying to doff my bp/w after a dive (I tend to pull my arm out of both straps instead of just the BC.
It's overkill if you only wear a little bit of weight, IMHO. I've actually managed to drop pretty much all of my lead for local dives after the water warms up, but I've actually used it for auxiliary pockets once or twice. It was great for our first real ocean trip when dialing in a new weight combo (different tanks than normal, salt water, anxiety/excitement of blue water). Easy to add/subtract in the pockets instead of trying to stuff weights in places they didn't belong (or didn't exist in my case). It actually makes the weights comfortable enough that you might forget to take them off after the dive now and them.