A recurring theme in the answers and comments that you have received so far is that the question is ill-posed since many (or perhaps all) non-existence claims can be recast as universal claims. While this is logically correct, I sympathize with your question and have thought about it a bit myself. While Liouville's theorem can correctly be phrased as the statement that bounded entire functions are constant, it really feels like a non-existence theorem: our experience with one variable calculus over the reals suggests that there ought to be a wealth of bounded complex differentiable functions, but in fact there aren't that many.
Here is why I think non-existence theorems are useful, and why I think it is useful to think about them as non-existence theorems. Mathematics is fundamentally all about starting with a huge untamed wilderness of possibilities and hacking our way through it until we have completely understood all possible behavior. This is why every area of mathematics has its own classification program which often motivates much of the research in that area; we are not content to just have a list of examples of a particular mathematical object unless we know that the list is in some sense comprehensive. Even when we can't classify some class of objects - such as finitely presented groups - we aren't satisfied until we can prove that we can't classify them.
A good illustration of what I mean is the classification of finite simple groups. At its heart, the result is a non-existence theorem: an important part of the proof was constructing all of the exceptional examples, but I think most of the 15000+ pages of the proof are dedicated to verifying that no other exceptional examples exist. By now I believe there actually are a few general theorems about finite groups that were proved using the classification theorem, but certainly not enough to justify the decades of labor that it took to complete the program. The reason that effort was worth it is because if it wasn't done then there would probably be a lot of group theorists out there right now up nights wondering if there is some crazy example of a finite simple group that nobody has thought of yet.
In the end mathematics is about building models for some sort of real-world phenomenon (even though it is difficult to trace some pieces of mathematics back to reality), and part of our job is to place constraints on what kinds of models are possible. It often pays off: the non-existence of bounded entire functions implies the fundamental theorem of algebra, a nontrivial existence result, and the non-existence of retractions from the disk to the circle implies the existence of a fixed point for any continuous self-map of the disk. I look at these results as examples of situations where we place enough constraints on a model to completely determine some of its behavior. This happens often enough to justify the overall philosophy in my mind.
