Hedgerow: drawbacks are you can't burn garbage in your stove, just good seasoned wood. A few more steps on firing up the stove vs non cat. Cat need replaced eventually. That is about it. Check out the thread I started about the EPA burn cycle there is a video showing the cat's effect on the smoke out the stack and thus burning the smoke.
Cat Stoves are typically for people serious about getting the most heat for the longest amount of time. Drawback is they are much more expensive that a standard stove - Although I think this will change in the next few years.
Also, they are typically much 'uglier' than other types of stoves. Although much nicer units are starting to come onto the market.
Finally, typically a horrible air-wash. ie. the glass gets dirty much faster than a typical stove and this is hard to fix on a Cat stove.
Someone had asked the question why would anyone not buy one instead of a standard stove? Price for one.. about twice as much as a standard stove. Since the average American wants to buy a stove at a big box store for less than a $1,000 this is quite a challenge.