So what is the point of a short-scale bass anyway? Is the main point a difference in tone, a difference in feel, etc...?
A fair question to ask.
There are differences in tone, as people have rightly pointed out they are hard to isolate. But they are there. And some, like the Dannos or Hofners, have a look and sound all their own. (IMO taken as a group they tend to be punchier, less sustain, tend to have slightly higher action, less tension and tend to be divy, but I wouldn't want to go to court on that because you'll find exceptions to every point and I've held and heard long scales with the same qualities, which is a long way to say what Pilgrim said better in his post.)
One reason for short scales is the one that prompted the original question, something that doesn't take up much room, for travel, tight spaces, ease of maneuvering.
Given some of the tiny spaces bands frequently are given set up in, I'm surprised you don't see more of them.
They can be very easy on the hands if you've had an injury or have other problems, and still deliver everything you want from a bass.
You can get all over the neck in a hurry if you want to.
Another reason, and one I see regularly when the discussion comes up, is that they are just plain fun to grab and play.
Everyone should own at least one. Just pick it up and enjoy.