See my answer here for a brief discussion of how points that are closed in one optic (rational solutions to a Diophantine equation, which are closed points on the variety over $\mathbb Q$ attached to the Diophantine equation) become non-closed in another optic (when we clear denominators and think of the Diophantine equation as defining a scheme over $\mathbb Z$).
In terms of rings (and connecting to Qiaochu's answer), under the natural map
$\mathbb Z[x_1,...,x_n] \to \mathbb Q[x_1,...,x_n]$, the preimage of maximal ideals are prime, but not maximal.
These examples may give impression that non-closed points are most important in arithmetic situations, but actually that is not the case. The ring $\mathbb C[t]$ behaves much like $\mathbb Z$, and so one can have the same discussion with $\mathbb Z$ and $\mathbb Q$ replaced by $\mathbb C[t]$ and $\mathbb C(t)$. Why would one do this?
Well, suppose you have an equation (like $y^2 = x^3 + t$) which you want to study, where you think of $t$ as a parameter. To study the generic behaviour of this equation, you can think of it as a variety over $\mathbb C(t)$. But suppose you want to study the geometry for one particular value of $t_0$ of $t$. Then you need to pass from $\mathbb C(t)$ to $\mathbb C[t]$, so that you can apply the
homomorphism $\mathbb C[t] \to \mathbb C$ given by $t \mapsto t_0$ (specialization at $t_0$). This is completely
analogous to the situation considered in my linked answer, of taking integral solutions to a a Diophantine equation and then reducing them mod $p$.
What is the upshot? Basically, any serious study of varieties in families (whether arithmetic families, i.e. schemes over $\mathbb Z$, or geometric families, i.e. parameterized families
of varieties) requires scheme-theoretic techniques and the consideration of non-closed points.
(Of course, serious such studies were made by the Italian geometers, by Lefschetz, by Igusa, by Shimura, and by many others before Grothendieck's invention of schemes, but the whole point of schemes is to clarify what came before and to give a precise and workable theory that encompasses all of the contexts considered in the "old days", and is also more systematic and more powerful than the older techniques.)