That's an interesting question! There's no real hard and fast answer. But as a general rule, high-end synthetics such as PolarGuard 3D or PrimaLoft offer the same insulation, ounce for ounce, as down rated to around 600 to 650 fill.
The Cat's Meow
The Cat's MeowFor instance, a typical synthetic-fill bag rated to 20 degrees, such as 's Cat's Meow ($169), weighs 2 pounds 10 ounces, virtually the same as the 600-fill down, 15-degree Blue Kazoo ($249), also from The North Face. 's 15-degree Helium ($369) uses 850-fill down and comes in about 4 ounces below the other two bags, despite its lower temp rating.
You see a similar differential in cold-weather bags. 's Lamina -30 synthetic-fill bag ($245) weighs a fairly hefty 5 pounds 9 ounces. ' -25 Ptarmigan, on the other hand, comes in at 3 pounds 15 ounces.
Of course, the Ptarmigan also costs a bunch more—$709 in the eVent shell version. And that's one difference between synthetic and down: down almost always costs more, especially in the more exotic high-fill bags. In my experience down also packs into a smaller space. And because down-fill bags typically are softer than synthetics, down bags “drape” better and so can seem a little warmer.
The real difference, however, is that a soaked synthetic bag will still retain about 60 percent or better of its insulating ability, and will dry our relatively quickly. Down actually can soak up quite a bit before its insulating capabilities are really degraded, but once down gets wet, you're cold. And a wet down bag takes a LONG time to dry out.
So I usually recommend synthetic bags where cost is an issue, or the camping is apt to be really wet. In all other cases I favor down. In my view, it's still measurable better in most ways.