This is a common question. Make this decision determined by your trip. Are you going to be traveling to a place where it rarely rains? Are weight and space a huge factor?
Down Bag Breakdown: Down bags are regarded as the lighter option for sleeping bags. One big plus to having a down bag is compressibility. If you are working with very little space, having a down bag can be a huge help. Ounce per ounce, down will be the warmer of the two fills. It will also keep its shape longer than a synthetic bag. The biggest downside to this fill is its extreme aversion to water. Once your down sleeping bag gets wet you are pretty much S.O.L. You are then. This is where a synthetic bag wins out. Also, down tends to be significantly more expensive than synthetic bags. For those on a budget, down may not be in the cards.
Synthetic Bag Breakdown: Synthetic bags have several pros over the down sleeping bag. With a lower price, and care instructions that do not include a list of do’s and don’ts, a synthetic fill bag can be a great alternative to down. Synthetic bags dry quickly. Synthetic fills hold their insulation, even when wet. No need to fret when your bag gets rained on. Leave it out to dry and you will be good to go by the next night. One downside to the synthetic bag, is that it does not compress as well as a down bag does. If space is a factor on a backpacking trip, a down bag might make more sense, unless you are going to a place with rain.