Why were algebraic geometers in the 19th Century thinking of
m-Spec as the set of points of an affine variety associated to the
ring whereas, sometime in the middle of the 20 Century, people started to think Spec was more appropriate as the "set of points".

What are advantages of the Spec approach? Specific theorems?

The basic reason in my mind for using Spec is because it makes the category of affine schemes equivalent to the category of commutative rings. This means that if you get confused about what's going on geometrically (which you will), you can fall back to working with the algebra. And if you have some awesome results in commutative algebra, they automagically become results in geometry.

There's another reason that Spec is more natural. First, I need to convince you that any kind of geometry should be done in LRS, the category of locally-ringed spaces. A locally-ringed space is a topological space with a sheaf of rings ("the sheaf of (admissible) functions on the space") such that the stalks are local rings. Why should the stalks be local rings? Because even if you generalize (or specialize) your notion of a function, you want to have the notion of a function vanishing at a point, and those functions that vanish at a point should be a very special (read: unique maximal) ideal in the stalk. Alternatively, the values of functions at points should be elements of fields; if the value is an element of some other kind of ring, then you're not really looking at a point.

Suppose you believe that geometry should be done in LRS. Then there is a very natural functor LRS→Ring given by (X,OX)→OX(X). It turns out that this functor has an adjoint: our hero Spec. For any locally ringed space X and any ring A, we have HomLRS(X,Spec(A))=HomRing(A,OX(X)) ... it may look a little funny because you're not used to contravariant functors being adjoints. This is another reason that spaces of the form Spec(A) (rather than mSpec(A)) are very special.

Exercise: what if you just worked in RS, the category of ringed spaces? What would your special collection of spaces be? Hint: it's really boring.

Edit: Since there doesn't seem to be much interest in my exercise, I'll just post the solution. The adjoint to the functor RS→Ring which takes a ringed space to global sections of the structure sheaf is the functor which takes a ring to the one point topological space, with structure sheaf equal to the ring.
