As others have mentioned, nilpotent elements show up (at least) in the structure rings of varieties counted with multiplicities. Why should we want to have such objects? I can think of at least three reasons:
- The concept of family is easier to deal with. For instance, in the context of schemes, it is easy to speak of a family of conics degenerating to a double line. If we replace the double line with the same line counted once, the family behaves more badly (it is not flat)
- As rings have fibered coproducts (tensor products), schemes have fibered products. This is a general construction with good categorical properties, and it generalizes a variety of contexts (fibers, intersections, pullbacks of vector bundles...). If you want to stick with varieties, this construction will not be available, as the tensor products of reduced rings can be non-reduced
- A particular non-reduced scheme $k[x]/(x^2)$ is very useful in deformation theory. In deformation theory you want to study a given map up to the first order (or maybe higher orders, so rings like $k[x]/(x^n)$ appear). The existence of non-reduced schemes allows you indentify such an object (a first order approximation to some map) with an actual map from $k[x]/(x^2)$. This is quite handy and simplifies many arguments.
One more reason to be happy in keeping schemes the way they are is the existence of the Quot scheme. This is a general construction due to Grothendieck which allows you to have schemes which parametrizes a manifold of objects: subschemes, morphisms and so on. Moreover, most other moduli space in use in algebraic geometry are constructed starting from a Quot scheme, typically as a GIT quotient.
There is no corresponding general existence theorem in the context of varieties, so moduli theorists would have a pretty hard time abandoning schemes. Of course often it happens that the relevant Quot schemes are actually varieties, but we do not know how to construct them directly. It is easier to produce something (a scheme) and then show that it is nice (a variety), than producing a nice object in one step.