Butane Stoves
The major advantage of butane stoves is simplicity. Essentially, they contain three parts: a canister of compressed butane gas, a valve and a burner. To light a butane stove, you flick your lighter and open the valve. The stoves are mercifully quiet, and there's no priming to worry about, no liquid fuel to spill and essentially no maintenance to do. I've used butane stoves almost exclusively since 1982, and never seen one break down. Butane stoves are not idiot-proof, but the average terminally confused backpacker has a hard time making them malfunction. By contrast, almost every backpacker can tell you a story about a white-gas stove that became cantankerous.
The main disadvantage of butane stoves is poor heat output in cold weather. Butane stoves only work if the cartridge is warm enough for the butane inside to vaporize into energetic little gas molecules that are eager to flow out the nozzle and burn. Normal butane condenses to a liquid at 31 degrees F. Stoves burning normal butane produce practically no heat below that temperature because very little butane gas flows out through the burner when the valve is opened. Fortunately, cartridges are now available for some butane stoves which contain a mixture of propane and butane. Propane liquefies at -44 degrees F, so the combined fuel works much better than normal butane in severe cold. MSR and other manufacturers offer stoves that burn iso-butane, a variety of butane that liquefies at 11 degrees F. These stoves also work better in the cold than normal butane ones. High altitude, with its lower air pressure, further increases the output of butane stoves compared to their performance at sea-level. Just as water boils (vaporizes) at a lower temperature as you go higher, so too do butane and its variants. A lower boiling point means more butane gas flows out at a given temperature. You'll get much better performance at 14,300 feet on McKinley at zero degrees, for example, than you will during a January cold snap in Arches National Park at 5,200 feet.
I now do nearly all of my backpacking in the spring, summer and fall in the United States. For me, the simplicity and ease of use of a butane stove, particularly now that the butane/propane cartridges are available, makes it the stove of choice. However, I grant you that using a butane stove successfully in the wintertime at normal altitudes, even with the new fuels, is a tedious process. If you plan to do a lot of cold-weather backpacking in the United States, buy a white-gas stove.
Adventurous backpackers planning to trek through Nepal, Patagonia or Outer Mongolia will find that white gas is generally unavailable outside the U.S, while stores carrying butane-filled Camping Gaz cartridges are surprisingly widespread (well, maybe not in Outer Mongolia). Kerosene is universally available, but the quality varies widely and it must be filtered carefully. Many stoves on the market today can be set up to burn either kerosene or white gas; several will also burn automotive gas. Airline regulations prohibit carrying fuel of any type on board, so you'll have to plan on obtaining it once you arrive.