So, what this pan does is actually very simple; the fins on the sides provide more surface area to catch the heat that slides up the side by convection forces when the pan sits on a gas burner. The "gas burner" part is incredibly important, as if you have an electric burner there will be negligible benefit, and maybe even a negative result. That extra surface area can bleed heat as well as it collects it. And since the pans are cast aluminum, if you have an induction cooktop they won't work at all.
So, let's say you have a gas burner, and one of these pans. Here's what I see as a potential issue. The walls of this pan will get hotter than they do when you use another more traditional type of pan. And that's not necessarily a problem, as long as you keep stirring. But that extra heat will tend to cause liquid at the edge/top of the contents of the pan (the meniscus) to heat far more aggressively. Which means that you will likely get a degree of crusting, scorching, etc...depending on what's in the pan, of course. Water? No problem, it's water. But if you're cooking a sauce, or making something like boxed risotto (not the real hardcore risotto, which requires constant stirring and so would not scorch) or some other grain, you may have some issues. They have a stockpot, which at first would seem like the ideal situation...except that if you're doing most things you would do with a classical stockpot (like making a large batch of stock or soup or stew) you may have MAJOR issues with that scorching.
I have to say...I have a gas cooktop, I cook a lot, I cook elaborately, we have a gas dryer, we have gas-fired heat in the winter. It's a decent-sized single family home. And my gas bill doesn't get high at all...average is a bit less than $50/month. I find it hard to imagine that these pots would make much of a difference in my gas consumption at all. Maybe if my cooktop were really wimpy, the speed of cooking would be nice...but isn't the better option just to get a better cooktop in that case? These pans don't help if you're using a skillet, or the oven (which would also probably be weak if the top burners of the stove are weak), and they cost quite a lot. It'd be cheaper to just upgrade the cooktop than replace all of your pans with this, and the results will be more controllable. I'd love a big pot to boil water for pasta that worked like this...but for every other application it seems to me that upgrading the range would be a better way to go.
But hey, that's just my two cents.