Possible downfalls for both types include:
Dial;
1. Chips getting into the teeth, causing the dial to 'skip'. This is less common now than years ago, but I still see it happening on occasion. If the environment is has small chips or lots of dust, this could be more common.
2. Not being able to set the dial to zero removes some abilities to measure in different ways (measuring the length of a pin, set to zero, putting the pin in a blind hole and measuring to get depth of hole for example).
Digital;
1. If they get wet, they must be completely dried before using (unless they are water proof--never used these so I don't know if they work or not).
2. They require a battery to operate, without the battery, the calipers are just a paper weight. Batteries typically last a year or more (mine have been going on 2 years now) so when it runs out usually a spare is not to be found.
As others have pointed out, if you buy a set, get a good quality set. I have to calibrate hand held instruments here at work, cheep models usually don't pass even when new (this may be my prejudice talking--but I like good quality instruments)