It seems to me that there are a lot of great answers here but I am not sure if they live up to the challenge of showing the usefulness of schemes through an elementary example.
Let me try my luck and risk the wrath of the MO crusaders.
Probably anyone having any kind of mathematical background have seen the classification of conics (a.k.a. plane quadrics) over $\mathbb R$. Perhaps still a large percentage of those who saw this early in their mathematical career wondered about the asymmetry involved in that you have the usual nice ones (ellipse, parabola, hyperbola), the reasonable degenerate one (two different lines, either intersecting or parallel), and then there are the weird ones (double line, point, empty set).
This last bunch, while clear from the proof seems odd and one might feel that they should not be considered. Now if one looks at them scheme theoretically, then it becomes completely clear how these fit into the same mold and how those "weird" ones are really the same as the "reasonable" degenerate ones. The double line especially is hard to explain without schemes, while the other two is really a consequence of $\mathbb R$ not being algebraically closed, although interestingly, the associated schemes actually kind of "see" the non-real points as well.
A more high brow version of the same idea is the fact that (say over $\mathbb C$) a family of elliptic curves (topologically tori) can degenerate to a nodal cubic curve (topologically a sphere with two points glued together) and a family of rational curves (topologically spheres) can also degenerate to the same object. This seems to lead to at least confusion if not to contradiction as it seems to indicate that a sphere can be continuously transformed into a torus. Looking at these families the proper way, that is, using schemes we see that the two degenerations are not producing the same scheme. The one coming from the rational curves will add an embedded point at the node while that embedded point does not appear in the family of elliptic curves.