in a lot of algebraic geometry books I've heard that working over $\mathrm{Spec}(A)$ is better than working over $\mathrm{Spmax}(A)$ in the case where you consider a variety over a non-agebraically closed field $k$. Of course, the prime spectrum suits better because of the functoriality, but in a lot of books dealing with arithmetic of curves (in affine charts, without sheaves) there are discussions and proofs of facts dealing with non-algebraically closed fields (even zeta functions) using only maximal ideals of reduced $k$-algebras (zero sets of polynomials).
Therefore I'm searching for a simple example where prime ideals are better than the maximal ideals. Furthermore I would like to know how should I think about non-maximal prime ideals (points that are not geometric). I think that it would be useful, too, if someone could explains what's the difference between $\mathrm{Spec}(k)$ and $\mathrm{Spec}(k')$.
Thanks in advance.
If $k$ is an algebraically closed field and $A=k[T_1,...,T_n]/\mathfrak p\,$ (with $\mathfrak p$ a prime ideal) is a finitely generated domain over $k$, you should think of the affine scheme $\mathrm{Spec}(A)$ as the variety $V=V(\mathfrak p)\subset \mathbb A^n(k) $ to which you adjoin formally a point $\eta_W $ for every irreducible subvariety $W\subset V$.
If $k$ is not algebraically closed the advantage of schemes over varieties is more convincing: the schemes $\mathrm{Spec}(\mathbb R[X,Y]/(X,Y))$ and $\mathrm{Spec}(\mathbb R[X,Y]/(X^2+Y^2))$ are completely different (they don't even have the same dimension) whereas the varieties $V(X,Y)\subset \mathbb R^2$ and $V(X^2+Y^2)\subset \mathbb R^2$ are both equal to a single point.
But the really convincing superiority of schemes over varieties is attested by the amazing results obtained through the use of scheme theory in the last half century by geometers and arithmeticians like Grothendieck, Hironaka, Deligne, Faltings, Wiles and a good proportion of Fields medalists and Abel prize recipients in that period.