What do we gain by considering spectra? There are two topologies which are known as the "Zariski topology" - one of them a topology on an algebraic set (henceforth a 'variety') $V$ in $A^n(k)$, and the other on $Spec(R)$ where $R$ is a ring.
(Let $k$ be algebraically closed throughout, and write $k(V)$ for the coordinate ring). I understand that the maximal ideals of $Spec(k(V))$ correspond (functorially) to points of $V$, that the 'inclusion' $V \to Spec(k(V))$ is a homeomorphism onto its image, and that elements of $Spec(k(V))$ in general correspond to irreducible components of $V$. I also understand that more generally we get a functor $R \to Spec(R)$.
However, it is not clear to me why we would want to topologise $Spec(k(V))$ at all, not what the topology really "means". My uncertainties are:

*

*The points in the space $Spec(k(V))$ do not all correspond to "points" in the usual geometric sense - some of them correspond to components of the variety V. It seems unorthodox to mix these different types of object.

*Do the extra points in $Spec(k(V))$ give us more information about the geometry of V? For instance, if $V$, $W$ are homeomorphic, are $Spec(k(V))$ and $Spec(k(W))$ homeomorphic?

*What idea are we trying to capture with the topology on $Spec(k(V))$? When should points and components lie in the same open/closed sets, and why?

It seems that the perspective of studying spectra rather than zero loci is fundamental to the modern viewpoint so I'd quite like to understand it.
 A: Thanks to @KReiser's comment linking to a related MO question, I now feel able to answer this question in full.
Question 1:
The key here seems to be that if $\mathfrak{p} \in Spec(k(V))$ is prime and not maximal, and is associated with an irreducible component $W \subseteq V$, then we don't think about $\mathfrak{p}$ as being identified with the component $W$,  but rather we think about it as a "generic" point which somehow lives everywhere in $W$ at the same time. Having access to these generic points is extremely useful and appears to be one of the main motivators for considering non-maximal prime ideals topologically. Proofs for generic points can be easily upgraded to proofs for all points (see Q2 below) and generic points are very helpful when changing the base field of a variety (e.g. from $\mathbb{Q}$ to $\mathbb{C}$).
Question 2:
Asking this question at the level of homeomorphism is really the wrong question to ask (and I'm not actually sure what the answer is). Instead, we should consider the question for locally ringed spaces. Then studying $V$ and studying $Spec(k(V)$ are in fact equivalent via the following:
Hartshorne, Proposition 2.6: 
The functor $$\textrm{Var} \to \textrm{Sch}$$ $$V \mapsto Spec(k(V))$$ from varieties over $k$ to schemes over $k$ is fully faithful, and in particular $V$, $W$ are isomorphic as varities iff $Spec(k(V))$, $Spec(k(W))$ are isomorphic as schemes over $k$. (The image of this functor is in fact the set of quasi-projective integral schemes over $k$ by Proposition 4.10).
In fact, this principle holds much more generally:
EGA IV, Proposition 10.9.6:
Define a Jacobson scheme to be a scheme where all of the rings in the structure sheaf are Jacobson rings, and let C be the category of such schemes. Define an ultra-affine scheme to be $MaxSpec(R)$ for some Jacobson ring $R$ with the induced local ring structure from $Spec(R)$ and define an ultrascheme to be a locally ringed space locally isomorphic to an ultra-affine scheme. Let $D$ be the category of ultraschemes. Then $C$, $D$ are equivalent categories via the forgetful functor which forgets the non-maximal ideals.
(This result makes a lot of intuitive sense: the Jacobson condition is exactly a condition for passing between prime ideals and maximal ideals)
So it seems that for a very large class of schemes, considering Spec is equivalent to considering MaxSpec, and that we consider Spec for reasons of convenience.
Question 3:
One way to get an explicit grasp on the topology on $Spec(k(V))$ is as follows: when $\mathfrak{m}$ corresponds to a classical point $P$ and $\mathfrak{p}$ to an irreducible component $W$ then $P$ lies in $W$ exactly when $\mathfrak{m}$ lies in the closure of $\mathfrak{p}$. (In fact I think the Zariski topology on $Spec(k(V))$ is the coarsest such topology). This topology is good for reasoning about generic points; if some property holds on closed sets and holds at a generic point p, then it holds everywhere in the closure of $\mathfrak{p}$ (classically: at all the points in the component corresponding to $\mathfrak{p}$).
A more abstract viewpoint is that for locally ringed spaces $(X,\mathcal{O}_X)$, one can define a functor $\textrm{LRS} \to \textrm{Ring}$, $X \mapsto$ $\mathcal{O}_X(X)$ and the functor $Spec: \textrm{Ring} \to \textrm{LRS}$ is adjoint to this functor, so represents the most general way to "go the other way" and get a locally ringed space from a ring.
