Do they work?
Whether or not "they" work depends on which "they" you refer to at any point. A good ab workout works. You're best off finding one made by someone you know is a good source of this kind of information.
It's important to be wary and not take advice from anyone who is fit. There are too many people out there flashing their abs, and trying to sell their "shortcut" program.
At the end of the day, if there was a documented shortcut to six-pack abs or whatever else, everyone would be using that shortcut, and it wouldn't be a shortcut. It would just be a method that everyone uses. There's that Catch-22.
Difference between these ab workouts and the usual weightlifting-based approach
Like any muscle, your abs will respond to the same methods of progressive overload. Adding more resistance, incorporating variety etc. But some muscles recover faster for various reasons, and can be trained more often than the usual "wait at least 48h".
The abs are a muscle group that is pretty much used throughout the day, and is a part of pretty much any substantial movement. As such, they tend to recover pretty quickly due to the frequency with which they're called upon to do work.
How often are they meant to be performed per week?
A good written routine will include that, but ab work can pretty much be done every day as long as you have the required equipment, and importantly, as long as you're feeling up to it. As with any "how often?" question, the answer is usually "as often as possible without causing you to feel chronic fatigue or pain".
As stated earlier, progression tends to follow the same rules as any other muscle. When it gets easy, add more resistance, and/or try a more advanced variation. And as always, whenever you ask yourself the question "should I do this variation, or that one?", the answer is "both". Variety is key.