Aerodynamically speaking, what difference does it's shape make?

I'm no aerodynamicist, so I can't offer hard numbers, just some thoughts:

As the stabilizer is a wing like any other wing, its shape will matter
as the shape of any wing.

Aerodynamics in coordinated flight is only part of the story.
Particularly spin recovery is crucial, which depends highly on the
airflow on the rudder, which in turn depends on a lot of things, e.g. on
the the shape of the horizontal stabilizer and the horizontal rudder
position.

Aesthetics certainly matters, too.