Carrera,
There are several issues with all screen cages that I find annoying at times, and occasionally bad for the health of the animals I keep. Most of these issues are indicative of the species I choose to work with and likely would not effect the average panther or veiled keeper.
First, screen cages are horrible insulators. I find it inconvenient and inefficient to heat and cool and entire room for the sole purpose of maintaining proper temperature ranges in a 2x2x4 cage or the like. I also find the rapid swings in temperature of open screen cages unacceptable. By using other types of materials for at least the sides and back you can provide some insulating properties to your habitat which will make creating microclimates much easier. It will also prevent the extreme rapid swing in temp, say when a door is opened or the air conditioner kicks on. These things will change the temp but the hope is that it will happen gradually enough that the chameleon can make choices about the gradient and react.
Second, maintaining humidity in an all screen enclosure is like trying to bail water from a boat with a spaghetti strainer. Sure you can tape shower curtains on your cage and other nonsense. Of course the more densely the cage is planted the better it will hold humidity. However solid walls on at least three sides and the bottom provide humidity gradients and again the cham can choose where it is most comfortable. For example the bottom corners of my cages hold humidity very well and when the heat rises you can be sure to find all of my chams down towards the back corners soaking up the humidity and staying cool.
Third, drainage. I do not like the look of a cage sitting on top of a Rubbermaid tub, or with an appliance pan shoved in the bottom. But more than that I do not find it sanitary. The water in that catch basin has passed through the potted plants and washed over fecal matter, crickets have drowned in it. It is a bacteria haven and unless emptied and bleached daily can cause problems. With the drain pan inside there is often gaps between the sides and the basin that the cham could get stuck in, feeders for sure, not to mention the chams access to the water. Even if you cover it in mesh do you want your cham trying to shoot into that dirty water to grab a drowning cricket. By using solid walls and bases I can plumb my enclosure, mine drain out side to a flower bed, this also allows me to mist for extreme periods of time if needed to simulate rainy seasons, hydrate sick chams, etc.
Next, aesthetics. This is big for me. I hate the idea of buying a beautiful chameleon, with great potential for color and size, only to stick it in an unsightly cage. I find screen cages difficult to hardscape and plant as well. I like to give my animals as many perches as possible of varying sized and textures, again in gradients so that they can choose. Same with planting density. While this can be accomplished in a screen cage I find it more difficult than when I can drill the walls and hot glue branches in place, mount planters for orchids and vine type plants, etc.
Last, injury. Some chameleons will rub their noses and heads on screen till it is raw. They likely do not recognize the screen as a barrier, like an iguana plunging headfirst into glass, and why should they it is not in their instinctive behavior set to recognize a man made transparent barrier. In addition larger chameleons will often climb screen and break off toes, causing infection, broken joints etc. These are just issues I would rather not deal with.
In addition solid wall enclosures provide more privacy and security to the chameleon. This in my opinion increases psychological well being.
So in short, ha, I find them unsuited to keeping my chameleons. I do however see there use for new hobbyist as convenient and the lesser of certain evils. If you do some research you will find I am not alone. On the other side of the pond keeping chameleons in terrariums is more common and accepted than in the US, and by most opinion they are way ahead of us in breeding efforts and chameleons health. This cannot solely be attributed to caging, but it must play some role.