Down is a better pound-for-pound insulator with a higher warmth-to-weight ratio than synthetics, and it fluffs up to its open (air-trapping) state more quickly and completely.
Down is a gossamer cocoon, light and airy compared to synthetics, although some people are allergic to down, which precludes its use.
With good care, high-quality down bags outlast synthetics because down clusters degrade much more slowly than synthetic filaments. The average down-filled bag might last 10 to 15 years, while synthetic filled bags typically are only good for three to six years (see Section 6).
Ease of carrying
Down is lighter and can be compressed to a smaller size (less bulky) than synthetic-filled bags.
Wet or humid conditions
Synthetics finally win a round. Down loses most of its insulating ability when wet and is nearly impossible to dry in the field. Synthetic bags can hold in warmth even when wet, and they dry much more quickly than down. To use down when it rains, you'll need a Gortex (or similar material) bivouac bag.
Synthetics are generally cheaper than down, but you need to factor in their shorter life when calculating your best option.
Most synthetic bags can be thrown into the washing machine, while fragile down is best cleaned professionally.