There are algorithms for determining the solvability in integers of certain classes of Diophantine equations. For example, we can deal quite easily with linear equations in arbitrarily many variables, and there are infinitely many of those. There is also an algorithm for quadratic Diophantine equations in arbitrarily many variables, though this is less obvious. The answer for cubics is not known. There is no general algorithm for determining the solvability of quartic equations in arbitrarily many variables.
There is a polynomial $P(w,x_1,x_2,x_n)$ with integer coefficients such that there is no algorithm that will determine, given input $w$, whether there are integers $x_1,x_2,\dots,x_n$ such that $P(w,x_1,x_2,\dots,x_n)=0$. As a consequence of this, there are infinitely many $w$ such that the equation has no solution, but the fact that it has no solution is not provable in first-order Peano arithmetic. A similar remark could be made about ZFC. I do not think that this result can be equated with "impossible to determine," but it goes some distance toward that.
The above results are very beautiful. But they do not necessarily have a direct connection with classical questions in Diophantine equations, such as questions abut elliptic curves. The work on Hilbert's $10$th problem in fact produced interesting number-theoretic results, and generated new number-theoretic problems. It has enriched Number Theory, and in no way diminished it.
