Plasma vs LED: Thickness and Weight
On average, an edge-lit LED TV is the thinnest, with only 1.8". Plasmas are not so bad anymore, about 2.5" deep in average.
Thickness is mostly for the appearance and impressing your friends. A very slim television looks a lot better in your living room than a bulky one. It is also easier to mount on the wall, because it will be closer to the wall.
You should not base your buying decision on only this factor, unless the esthetic of your television is very important to you and how it fits into your room. Consider the thinness a bonus if you are buying an LED TV.
A plasma television is heavy. Each cell of gas representing a color of a pixel have its own chamber, between two panels of glass. The glass must be strong due to the difference of pressure between the ambient air and the gas chamber. An LCD screen is lighter because it does not need panels of glass, it can be just plastic.
Like the thickness, the weight of the television should not be a main factor when choosing your television. However, a lighter TV means an easier time moving it around or hanging it on the wall. The lighter it is, the less likely you will break it.