To me, if there is a characteristic sound of horns, it's simply the dynamics. It's about hearing things sound live and real for the first time. There are a number of other "sounds" that you encounter, some that aren't so good that drive people away, but at it's core, there is a unique attribute in the sound that comes from extremely low power compression and when you get the bug, there aren't cone/dome speakers that can approach them. It's all about minimizing the bad stuff or reducing it to a tolerable level in many cases.
I used to be a devout horn hater until I heard Unity Horns and, later, modern waveguide based horn systems. I then heard horn systems with the smoothness of a good dome tweeter with all of the dynamic benefits, to boot.