Stick Frame Roof
This is the traditional way to build a roof, which is to say using standard dimensional lumber and doing all the work on site. Due to cost issues of large dimension lumber, builders will sometimes substitute I-joist for solid lumber. The principles are the same, although I-joists often need additional bracing because they are very floppy.
The advantage to stick frame roofs is that if you make the roof pitch steep enough, some of the space under the roof is useable either as rooms or storage, although obviously that means for lower slope roofs there is no advantage. Attic space is particularly useful in small lots in dense urban setting where the buildings footprint is limited by the small lot.
In most cases using a factory built truss is significantly cheaper than a stick frame roof. Material wise, stick frame roofs generally require larger dimension lumber than the equivalent truss roof, although the difference isn't a big if you use attic trusses.
As with walls, a prime consideration on framing method is whether or not insulation is going in the cavity or not, and how much insulation is needed because the space required for insulation is sometimes greater than the space created due to structural concerns (eg you want 12" or even 16" of insulation, but the cavity is only 8" or 10"). Due to this space requirement it is generally much harder to insulate a roof than it is the attic floor because you may have to add additional structure to make a large enough cavity for the desired level of insulation. This of course assumes you want a very large amount of insulation, which given that you're reading a green building website is hopefully a good assumption.
The alternative for insulating sloped roofs is to add exterior rigid insulation As with walls the key to taking this route is to make sure there is enough insulation on the exterior of the roof so that the sheathing underneath it stays above the dew point. This can be a little more challenging in roofs because in some climates (cold dry in particular) the roof can get quite a bit colder at night than air temperature. In the opposite condition (hot humid), the situation is easier because most exterior rigid insulation is not very vapor permeable.
Vented versus Unvented Roofs
If you're going to put insulation in between the roof rafters, you have do consider whether you need ventilation or not. The conventional wisdom was that you always do, but builders are discovering that its not the case. As with walls, if the roof sheathing is cold and vapor can get at it, you have a problem, and putting a vent space between the insulation and the sheathing is the solution. The alternative then is obviously either to stop the vapor movement, or to keep the sheathing warm enough so that its above the dew point. In this discussion the assumption is that the insulation being installed is vapor permeable, ie any fiberglass, cellulose, rock wool or open cell foam. If the whole cavity is filled with non-permeable closed cell foam, then problem solved and you don't need venting1.
Option one, venting: in this case the framing material will be taller than the insulation in it, leaving a gap at the top between the insulation and the sheathing. The gap is then vented with soffit vents and continuous ridge vents (or equivalent) so that any moisture that gets thru the insulation dries out due to air movement: at least that's the theory. This air, of course is at outside temperature, so if you're using venting, there is no reason to put rigid insulation above the sheathing as all you'd be doing is insulation outside air temperature from itself.
Some builders have started to question how well venting actually works. Roof venting works by stack effect, ie the air in the vent is heated by the sun and rises to go out the top. Clearly it doesn't work when the sun doesn't shine and it doesn't necessarily work when the wind is fighting against it, and it also doesn't work well if the roof line doesn't provide a more-or-less straight shot since any bend is resistance to air movement, especially considering the vent channel is typically only 1". Also, the air moving in the vent isn't necessarily dry. That said, as long as the stack effect works often enough and the air moving thru it is dry enough, venting works.2
Option two, stop the moisture: there are various ways you can do this. You use PVA paint, which is a vapor barrier, you could install poly sheeting under the drywall. or you could spray a couple of inches of closed cell foam against the sheathing. The problem with PVA paint and poly sheeting is that most moisture transport is generally due to moist air movement, not vapor, so unless the ceiling has no penetrations (ie no vents, no ducts, no plumbing, no fire sprinklers, no can lights, no electric boxes of any kind. To make it clear, to be safe it means there is not a single hole in the ceiling drywall.) there is some reasonable change that the hole is either not sealed well or that the seal will fail over time. This is a much bigger problem than in walls because warm air want to go up thru the ceiling much more than it wants to go thru walls. Obviously you can seal the penetrations really well, but how long will that seal last given that if your solution to avoiding roof sheathing rot is to stop moisture, you're betting the longevity of your roof on these seals. That leaves closed cell spray foam, which has its own set of environmental problems, but if you put enough in to be an air barrier, you're probably fine an if its also a vapor barrier, you're sure your roof won't rot. Note that open cell spray foam is vapor permeable, but a good air barrier, so although it although it solves the main problem (moist air movement) there is enough extra drive on a roof that its probably not that safe to use. If your sheathing is plywood (which is a bit more permeable than OSB), you might be able to put a vent channel above the sheathing and still use open cell foam, but the emphasis here is on might since your now counting on the moisture moving thru the sheathing and then drying in the vent.
Option three, keep the sheathing warm: this means installing some kind of rigid insulation on top of the sheathing. This often means installing a second layer of sheathing on top of the rigid so you have a something to attach the roofing to. On the roof it probably generally doesn't matter whether the rigid is vapor permeable or not, but if you use a tile roof over vapor permeable insulation on an air conditioned building..well that might not be a good idea. The amount of rigid insulation you need depends on your climate and how much insulation is below it; as with walls the idea is to keep the sheathing warmer than the dew point all the time
Summary: The big advantage is that you get usable attic space, but the downside to doing so is having to use more lumber and an increased difficulty in insulating under the roof.
Notes
1: That doesn't mean the building department will agree; in fact they probably won't.
2: See http://www.jlconline.com/insulation/roof-ventilation-update_2.aspx and http://www.greenbuildingadvisor.com/blogs/dept/musings/how-build-insulated-cathedral-ceiling for more detail on roof venting.