The overall point of a Watchdog is to ensure that the firmware is executing as expected. The theory is that if your firmware can periodically kick the watchdog, then the other functions it is responsible for are also happening.
From a system design, they're the last level of fail safe. It's basically saying "we don't know what the system is doing, because it's not able to kick the watchdog. So, reset the device and hope the problem goes away."
They can protect you from accidental infinite loops, stack corruptions, RAM bit twiddles, etc.
A Windowed Watchdog is a better solution than a single-sided Watchdog as the window can protect against more things... For example, with a single-sided, if the loop you're stuck in includes the watchdog kick, you'd never know you had a problem. For a Windowed Watchdog, you have a better chance of resetting due to the likelyhood of kicking too fast...
So, to answer your question. You'd use a Windowed Watchdog any time you wanted to be reasonably sure that the firmware is doing what it is supposed to, or to fall back to a safe state if it's not. They are generally focused on in safety systems, but all embedded devices can benefit from their use. (For example, a house thermostat is not considered a safety-critical system, however if it completely locks up and requires someone to remove the batteries to restart it that would be an annoyance.)