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.