Drive belts are very useful, since they're powering various important auxiliaries.

Timing belts on the other hand are powering the camshaft(s)... and if there is a hell, that's where this idea was born (the Glas factory in Bavaria, using the first timing belt in 1962). In the 70s, those belts became fashionable to a point that even Porsche used one in the 928. 

Sometimes I would like to know how many engines suffered massive mechanical damage because a timing belt went bald, or just broke, and how much the vehicle owners had to pay for the repairs. Plus the amount of money paid out for maintenance replacement. That should be a tidy sum.

Yes, I'm a bit touchy about timing belts since the engine in my car was originally developed in the 70s and has one of those hellish devices. Next year another 1100 euros will go down the drain, because the replacement interval has been shortened a while ago, from 6 years or 100.000 km to 4 years and 60.000 km. (I like the confidence expressed by this.) 

Of course the cost for repairs if the belt breaks within this interval has to come out of the owner's pocket too.Downsides of timing belts: cost and reliability. Replacements are just a part of it. 

Since until very recently (Ford 1.0 liter Eco-boost) timing belts can't run in oil, they have to be outside the engine block. That requires a casing to protect the belt against dirt, additional shaft sealings, it's adding to the overall engine length. It's rather telling that you won't find timing belts in aircraft engines.

Benefits: much to my bitter entertainment the German Wikipedia article has a list, which partly is countered by the following negative list. One the plus points is "does not require extra lubrication", and on the minus list you find "higher wear and tear because it's not lubricated". 

Another instance is "installed outside of the engine block, easier to replace", which is opposed by "higher cost for replacing" due to lots of extra work to actually reach it. In case of my Alfa, the set amount of time for a belt replacement is 6,5 hours (plus another half hour for the water pump, call that "preventive maintenance".

I can hear the question: "if you hate the timing belts so much, why do you drive a car with one?" It's not rational. I love the car, the noise it makes, the way it looks, how it drives, despite some taxing drawbacks. It already wasn't rational to buy it. I know that very well...