With no disrespect intended for the previous answers, they include some outdated and slightly incorrect information.
Like everything else, the answer is: 'It Depends.' Each type has its own advantages and disadvantages.
Screw drives are "direct drive" openers, sort of. To say that they contain fewer moving or mechanical parts is incorrect, and of the three, they tend to be the noisiest. People who have always used a screw-drive are understandably loyal to this type, and they are still sold today, but the industry as a whole is moving away from screw driven machines.
Chain driven openers are the middle ground as far as noise and vibration are concerned, and require very little upkeep - contrary to popular belief, regular lubrication of the chain doesn't make it last longer.
The belt drive openers tend to be the most expensive models, but they are also the smoothest and quietest type of opener out there. Check the warranty before you buy - most belt drive openers have at least a 10 year, and in many cases a lifetime warranty for belt replacement. The belts do not break or stretch (popular belief, but untrue). The belts are constructed in much the same way as a fan belt on an automobile, with steel reinforcement running the length of the belt. Opener companies aren't being generous by offering long warranties for their belts - they do it because they know that the belt will not break.