Wood is quieter off the bow. It's available in the widest range of spines and weights, making it easier to match to your bow if you already know what it requires. It's better looking than carbon or aluminum IMO. Of course, those of us who shoot wood also think it ties us more closely to times past and the spirit of archery. As others have said, wood just has soul. Also makes a good fire starter in emergency situations.
Cons. Good wood arrows are more difficult to build than aluminum or carbon arrows. It takes good shafting and good craftsmanship to build good wood arrows. There's a lot of junk out there -- bad grain, severely mismatched spine and weight, etc. And a lot of folks don't have the patience, skill or know how to build good wood arrows.
If you factor in the time required to make them, cost of quality raw material, the finishes needed to seal them, and the equipment needed to make them -- spine tester, grain scale, sanding disk taper tool, etc. -- good wood arrows will cost just as much as aluminum or carbon. In fact, a lot of junk wood arrows also cost just as much.
The bad wrap that some lay on wood arrows can almost always be traced to poorly made arrows, mismatched in spine and/or weight and/or having crooked nock and/or point tapers and thus nocks and points that are not aligned properly.
I've been shooting wood for about 50 years, and i spend very little time straightening arrows. I start with premium quality shafting, which is quite straight to begin with, straighten those shafts that need it and seal them. That's it. Unless I sit on them, try to yank them out of a target sideways or have a critter fall on them, they stay straight.