What's the advantages and disadvantages of each?

Is one more tolerant than the other of being stored in a stuff sack?

Sleeping in a wet bag does suck.  
The difference is, down (typically) loses almost all of its insulation abilities when it is wet, while synthetics typically do not.  
That can mean the difference between wet and uncomfortable vs. wet and dead.  

The con of most synthetic bags is the weight/compressability .  
The con of most down bags is the price.  The ones that have down which is treated to resist absorbing water are even pricier.  

Personally, if I'm going to an area that has high humidity and lots of rain; I'm taking my synthetic bag.  
If I'm camping in a dry region with little rain, the down bag works just fine.

A quality down bag will last a lifetime, stuff smaller, and provide better warmth/weight than any synthetic bag.

I've seen a bivy sack flood with icy water from an impromptu waterfall one night on the side of a remote Grade VI climb - hard to conjure a worse soaking - but, we had that down bag back in service by the following evening.  It was a long night, but it would have been that way regardless of what material the bag was made of.