Another answer to your question is provided by Mumford's book "Lectures on curves on an algebraic surface". This book explains Grothendieck's proof (via schemes) of the fact that for a sufficiently positive curve $C$ on a smooth projective surface $S$, the algebraic equivalence class of $C$ modulo the linear equivalence class of $C$ has maximal possible dimension, namely the dimension of the Picard variety of $S$.

According to Mumford's introduction, the only known earlier proofs were analytic in nature.

Mumford's book is really excellent by the way; it is not a substitute for Hartshorne or any other introductory textbook, but is a wonderful "second course", which introduces ideas such as the Hilbert and Picard schemes, some deformation theoretic ideas, and related techniques, and (perhaps more than Hartshorne) really shows how you can use all of Grothendieck's new methods to actually do something!

Of course, this is not an elementary problem in the sense of your question, but in thinking about algebraic geometry and its foundations, its worth remembering that algebraic geometry already had an extremely rich history by the time Grothendieck introduced schemes, and so the big outstanding problems (of which there were definitely many!) that demanded the introduction of this new technology were not simple ones (hence it's not so easy to justify the introduction of schemes with one or two very simple examples). (If one wants a truly simple example, one can just discuss how the size of a fibre under a map of projective curves is constant, provided that one counts the size of the fibre using its scheme structure, i.e. taking into account the nilpotents that appear at ramified points. But I don't know how compelling this example would be to non-algebraic geometers; it still may seem more like convenient book-keeping than a genuinely important new technique if you can't demonstrate it's utility through some specific application.)