I've shot pretty much everything, from cutting my own arrow stock from ash and douglas fir and dowelling it myself to middle-of-the-road carbons (I just won't pay $300 a dozen for high end ones).
Carbon absolutely recovers faster than wood. Carbon also noodles less when it hits the target. There are videos around that show both of those phenomena if you want to look.
As for variations in spine, carbons tend to differ in spine around the shaft, just the same as wood. If you want a truly matched set of arrows, you spend the five minutes to rotate them in a spine tester and mark them when you get the spine you want.
Having said all that, both kill critters just fine. If you want to spend the extra time to get a quality set of woodies exactly matched, you can come very, very close to the performance of a carbon arrow.
Me, I'd rather shoot these days, I shoot carbon.