Yes, down is warmer than synthetic. Down is the soft, fluffy, and lofty stuff under the feathers. Considered as nature’s insulator, down (made from duck plumage, not feathers) it is effective in trapping air and body heat, while still being breathable. Hence, it wicks unwanted moisture away from your body.
The effectiveness of down can be measured through “fill power”; the more it lofts, the higher quality it is. If you take one ounce of 800-fill down, it will supposedly take up 800 cubic inches; one ounce of 900-fill down will take up 900 cubic inches, etc.
On the flip side, synthetic sleeping bags are typically made from polyester. Most manufacturers use either short-staple fills or continuous-filament fills. Short-staple fills mimic the down’s structure and it compresses just as well, but it does not last long. Continuous-filament fills are more durable than its counterpart, but you would have a hard time compressing it for storage.
Another point is that synthetic sleeping bags are less expensive and is more hypoallergenic. However, synthetic does not keep you warm as efficiently as down does.