Well, the main advatages of down are weight and pack size. For a given level of insulation, down will weigh
much less and pack
much smaller. However, it's pretty much useless if it gets wet (or even damp) and is difficult to dry. It also needs to be washed very carefully and is a real nightmare to handle during construction. Synthetics are also cheaper.
Many, even most, people seem to prefer synthetic, but having recently bought a down sleeping bag, I'm a convert. The difference in packed size in particular is amazing - you just wouldn't believe how small a space down can be packed into if you haven't seen it.
Here's a photo of my down bag, packed, with a waterbottle for scale. OK, it's only a 2-season bag, but still... It's tiny. And only 480g.
My design goal is a 2-3 season underquilt weighing in under 600g, and packing down to no bigger than my hammock / net combo - about 6L.