I agree with John Hughes; this is an interesting question. It is made more interesting by the observation that we often do answer questions like "Solve $f(x) = 0$" with something like $\{x \mid g(x) = 0\}$, which seems on the face of it like answering one question with another. Typically, of course, $g(x)$ is simpler in some sense than $f(x)$: For instance, we might answer
$$
\text{Solve for $x \in \mathbb{N}$ in }8x+1-y^2 = 0, y \in \mathbb{N}
$$
with
$$
x \in \left\{\left.\frac{n(n+1)}{2} \,\,\right|\, n \in \mathbb{N}\right\}
$$
where we might think of $x-n(n+1)/2$ as the equation $g(x)$ that characterizes the solutions to $f(x) = 8x+1-y^2 = 0$ for $y \in \mathbb{N}$. (There are better examples; this is just the one that comes to mind.)
What makes this characterization of $x$ simpler or better than the one that is posed as the problem? The explanation that I come up with is that we agree that—there is consensus that—the solution is a more immediately transparent description of the $x$ that solve the problem than the problem itself is. In other words, mathematics (like science in this regard) is a social activity, with (often unspoken) agreements about what constitutes progress toward a more primitive characterization of a mathematical object.
EDIT (2024-02-01): Sometimes, as in this case, at least part of the distinction can be made explicit like this: In the former case, the most straightforward and direct way to find all solutions is a loop that, for all you know, is unbounded; it's not clear there are always more solutions. In the latter case, the most straightforward and direct way is simple—simply run through all natural numbers and plug them into a finite expression. But it's not always that clear-cut.
When we begin, as students, we get used to the idea of solutions being concrete, like the number $4$ for $3x-12 = 0$. Later, we apprehend that mathematics is a big web or network of relationships, and solutions often merely move from an expression that is less simple or transparent to one that is more simple or transparent. What it is that simplicity or transparency actually represent is not usually made explicit, and I'm not sure that it can be made explicit in any universal way.