Down is lighter than synthetic fibers and easier to compress. That's why down sleeping bags take up less space. They are also more breathable than synthetic sleeping bags.
On the other hand, synthetic sleeping bags are better than down ones when it comes to moisture.
Down sleeping bags are more durable - if they're washed and handled with care, they can last to up to 20 years, where synthetic sleeping bags usually last about 5 years. The durability depends, of course on the quality and the usage frequency.
Down sleeping bags are more expensive than synthetic fiber sleeping bags.