Like everything else, the answer is: It Depends. Each type has their own advantages and disadvantages.
Screw drives are "direct drive" openers, sort of. To say that they contain fewer moving or mechanical parts in incorrect, and of the three, they tend to be the noisiest. People that have always used a screw-drive are understandably loyal to this type, but the majority of openers sold today is moving away from screw drives.
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.