The common idea is that, on one hand we have "I don't have X", on the other hand we have "I have X, but X has some its quality equal to zero, making it to behave the same way as if it didn't exist", where X can be some object or property of mine.
Other examples. "I don't have speed"(because I don't move) VS "I have speed, but it's equal to zero"
"a point doesn't have length, area or volume" VS "a point has length, area and volume, but all of them are equal to zero"
"I don't have speed" does not mean the same thing as "I have speed, but it's equal to zero." The latter means that your position does not change in time (i.e. that you don't move), while the former means that your position is not given by a differentiable function of time. Speed is defined as the magnitude of the derivative of position with respect to time. There is a difference between saying this is equal to zero and saying it doesn't exist.
"a point doesn't have length, area or volume" VS "a point has length, area and volume, but all of them are equal to zero"
In general, "length," "area," and "volume" are defined as the 1-, 2-, and 3-dimensional Lebesgue measures, respectively. Since all of these are defined for a single point and equal to zero for that, it would be incorrect to say "a point doesn't have length, area or volume," unless you are in some context in which those terms are defined differently.
Getting back to your original question about horns, since we have not given a precise definition of "horns," the point about zero volume is irrelevant, since it may be the case that the significance of these horns does not depend on their volume. For instance, let's say you have horns that have a hole in them, and we are interested in whether the space around your horns (the complement of your horns) is simply connected. The answer to this question would be yes if you have no horns, while it would be no if you did, even if the horns had zero volume. Thus, we are comparing the statements "I have no horns" and "I have horns," which are clearly different.
Is there any philosophical difference between "I have no horns" and "I have horns, but they have zero volume"?
There is an enormous philosophical difference between these two statements but it has zero volume!
One difference is that "I have no horns" has existential import whereas "I have horns, but they have zero volume" does not.
Further, the former is either true or false. The latter is neither: I have one billion dollars, but they have zero value. Nobody can tell you whether this is true or false.
Thus, the difference is logical, with implications for the philosophy of logic.
One important distinction of this kind arises in statistics (and is closely related to measure theory, as mentioned in Sandejo's answer). Consider a random variable $X$ that's uniformly distributed over $[0, 1]$; i.e., $X$ is equally likely to take the value of any real number between $0$ and $1$. The probability of $X$ taking on any specific number in that range, such as $\tfrac{1}{2}$, is 0. We say that it's "almost sure" that $X ≠ \tfrac{1}{2}$. But $X$ still can, and must, take on one of these values. By contrast, it can't be 2, so $X ≠ 2$ with no qualifications.
