In addition to the fungi and bacteria attacking the wood when a tree dies as Pat described, the wood starts to dry out......in an uncontrollable fashion. When the trunk loses moisture content, it shrinks radially more than in other dimensions. Cracks and fissures open up that degrade the wood for use as a bow.
But, when a living tree is harvested and staves or billets are split out promptly, you can seal them to control moisture loss. The environments you store them in can also help control moisture loss in a manner that will yield good, seasoned wood from which to build a bow. You can control or eliminate the problems caused by radial shrinkage.
And, a dead, standing tree will often attract a host of insects which bore into the wood and render it useless for bow construction.
So, it's a huge gamble to harvest standing dead timber for bow building, at least not in our eastern forest. Maybe once in a while you could get lucky, but it's usually not worth the effort. That's my 2¢ worth.