Okay, first off, I love sponge filters. I'm a total convert from HOB's. But, other people feel differently, so part of it is just personal preference. :) There's no right or wrong answer. :)
-increased surface area means it's easier to cycle small tanks (sometimes the only way possible to cycle some small tanks), increased surface area is nice for bigger tanks too, and I think my cycles are stronger with my sponge filters...but that's just my limited experience, not necessarily universal.
-Smaller and easier to put into tanks because the only thing coming out of the tank is a small airline. This makes it much easier to put on hoods/versa tops/glass or plexi lids/ etc.
I like my tanks to be very minimal and tidy, so I like clear plexi lids and limited numbers of cords, filters, lines, whatever coming out of them. I try to position my heaters so that the cords come out at corners and are then secured under the bottom of the tank, and not having a big black HOB sticking out the back really improved the aesthetic for me. :) Just a single clear-airline is easy to hide and very easy to "lid" around. :)
-easier to hide because they can be covered in plants or put behind ornaments in a corner (one of mine is actually completely hidden in the tank.)
Also, if you make your own, you can use clear tubing for big central tube and for the airline tubing, and you can use whatever color sponge you want, which also makes it easy to camouflage them. I've found that a light blue sponge is one of the easiest to hide in most tanks because it blends in.
-increased oxygenation of the water if you baffle the living daylights out of a HOB. If you don't baffle the HOB's to the point of almost no water movement this is a moot point though. Since water movement=oxygen, the small bubbles of a sponge filter provide all the O2 without so much water disturbance.
-no baffling!! :D Also, easier on the fish because there's no current, and completely safe, they cannot get hurt on these things. No intake sponges, no outflow sponges, no torn fins, no getting sucked into the filter or pushed around by the outflow...no trying to figure out how to tone them down with water bottles or sponges or water levels.
- your friend is right, the mechanical filtration is really poor, almost nonexistent. But, because we're dealing with small tanks with limited fish loads in them anyway...I don't really consider this to be a con. When I vacuum my gravel once a week or every two weeks, I cannot tell any difference in the cleanliness of my tank from HOB or the Sponge. Also, when you baffle the intake on the HOB because you really must to keep the betta safe, you're limiting the HOB's ability to mechanically filter *anyway*.
If we were talking goldfish or some other large fish with lots of very visible waste, then that would be different...but bettas are small and clean, so it takes a LOT of time before tanks start to look nasty.
- No chemical filtration. You cannot use carbon in a sponge filter. (Well, I saw one guy who was trying it, but I doubt it was really doing much. The water has to flow over the carbon to really be cleaned by it.) But, this is only important if you need to clean something out of your tank...like meds or a pollutant. Since you really almost never want to treat your show tank anyway (meds in QT containers)...this is, again, not something I worry about.
I do keep a couple of HOB's from before around and some loose carbon just in case, but for day-to-day I use my sponge filters. :)
I hope that helps some! :)