It does to me, especially in terms on flexibility and ease of fretting, vibrato (which I use a fair share of), and bends, and to a lesser degree tone as well.
Personally I prefer light gauge, too thick with too much tension and you loose sustain, snap and bite, and since I play short scale the E string if too thick with too much tension it becomes too inflexible and have almost no sutain and becomes really thuddy.
What many don't realize, is that strings with the same tension and of the same thickness, if that was possible, will feel considerably stiffer the shorter the scale, as in being less flexible, which leads to the common misconception that short string basses needs higher gauge strings than you would use on a regular 34" scale bass.
This is basic physics, take a stick and extend it like a bridge between two raised points, the further apart the two points is the less force will be required to bend it on the middle, as in becoming more flexible, the closer distance between the two raised points the more force will be required to bend the same stick on the middle, as in becoming less flexible.
That's one of the main reasons why you typically will use strings with much much less tension on a guitar, the much shorter scale simply makes the strings still be sufficiently stiff to be playable, whereas the same tension on a 34" scale bass would become a flop fest.
In my opinion people generally will use too thick gauged strings on their short scales, resulting in a thuddy dead tone, for better tone they should use lighter gauged strings and instead adjust there picking technique to pick lighter.
I'd claim that a gauge .100 - .080 - .060 .045 string set would be just about ideal for a 30" scale bass, and actually I even personally prefer a .095 - .075 - .055 - .040 set, which, despite having less tension, in my experience still will feel less floppy than if put on a 34" scale bass, because of the increased relative stiffness that comes with the shorter string span of a 30" scale bass.