Not a stabilizing expert, but here is how I understand it:
The stabilizing process draws resin inside the blank via vacuum pressure. When the resin cures, the wood will be much more stable, will likely turn a little better, and should sand and finish well. Some prefer stabilized blanks for everything for the reasons mentioned above, but it can add a lot of time and cost to the project.
Stabilizing is done once the wood is dry. I've even heard of some people putting dry blanks in an oven before stabilizing to lower the moisture content a little more. The need for stabilizing wood depends more on the structure of the wood, rather than how it was dried. Burls are likely candidates for stabilizing, because often dried burls (at least the ones I've worked with) can feel kinda light and airy, and all of the inclusions and crazy grain structure can cause the blank to crack/break while turning. Another likely candidate is spalted wood. The black line spalting happens when a fungus gets in the wood and it is decaying. There is a fine line between when the spalting looks beautiful, and when it is too far gone to be much good for turning. Stabilizing the blanks is used to give those soft/punky blanks the stability they need to be turned and finished well.
This is how I understand things, again not an expert on the subject. Hopefully this helps you some, and someone else will come along that can give you more firsthand knowledge on the subject.
