I remember an airplane called the Stark Shark a few years back that had a very thick horizontal tail. According to the designer it had a nice soft pitch feel. As I was working for a living at the time I did some looking in the available literature and did discover that there was a benefit to effectiveness if everything else is kept the same, just thickness changed.

When you throw in varying area, elevator size, etc. the difference can get lost. Control deflections of 40 degrees when the elevator is half the horizontal size pretty well negates any differences. However I have always felt that the airfoiled tail is worth while on anything bigger than a .15 size airplane where precision is desired. I believe it is going to be the same on the vertical tail as well.

We use flat surfaces on the small foamys because they weight nothing and are overpowered (at least the fun good ones are) so arn't a good example of what an ideal aerodynamic machine should be. At the lower Reynolds numbers the flat plates arn't that bad. Look at a butterfly. But for anything that has a reasonable Reynolds number the drag, etc. of the airfoiled surface is better.

You can always point to some full scale machines and models also that have flat tails that work fine, however it doesn't mean that it is the best way to go.

And as has been mentioned, the structural strength for weight is better with a thick tail.