What situation would you use flexible termination ceramic capacitors?
Flexible-termination caps are commonly used in automotive applications, where vibration causes the board to flex back-and forth a small amount repeatedly. Normal components in this application can sprout fatigue stress-fractures that gradually enlarge over time until they fail, which in some automotive applications can have catastrophic effects.

It is very hard to completely remove vibration-caused flex without massively supporting a PCB, and that is very expensive, even compared to fancy caps like the Kemet ones linked.

As tronixstuff mentioned, they are also commonly used in situation where the environment is fairly extreme, as there is no way to completely prevent thermal stresses, since the different aspects of a PCB have different coefficients of expansion as the temperature changes.

Basically, they're for high-reliability/extreme-environment applications.