Ok, in my extremely limited experience...
Wool batting is warmer and lighter, but very expensive to buy (well, it's expensive here in Australia - not sure if the prices are perhaps more reasonable in the US?). It has a nice almost fluffy 'loft' that makes the finished quilt a little more cloud-like (I hope that makes sense) while still being light.
I've used cotton/bamboo batting for several quilts. Cheaper, but still good quality and very nice to work with, sits nice and flat as a finished quilt. I like the way it sits fairly snugly around us in bed - a bit heavier and a bit flatter perhaps? But not as warm.
My most recent quilt is my favourite yet but a lot more bulky. I used an old wool blanket I had lying around the house. Ugliest thing you'll ever see - I never used it and will absolutely not miss it as a purple blanket (Yukky purple, not a nice purple). I added cotton/bamboo batting as a layer with it as well. So it goes 'quilt top, cotton/bamboo layer, old purple wool blanket layer, fabric backing layer' - so there one more layer in there than you'd usually get. I used the spray stuff between all the layers to keep them together while I did the simple stitch in the ditch quilting. It was heavy but so worth it.
It's fairly bulky when you fold it up, but quite nice and heavy on you in bed and the warmest of the quilts I've made so far.
It was made entirely of scraps, and I've done it specifically for my camper (when it finally arrives!). I'm pretty happy with it but bear in mind I won't be folding it up all that often. It will live on the bed. I wouldn't suggest this if your quilt needs to be folded up and put away in a cupboard anytime soon!