Why is the prime spectrum not useful in non-archimedean analytic geometry? This semester I am attending a reading seminar on non-archimedean analytic geometry (a subject I know nothing about), roughly following the notes of Conrad. 
Reading Conrad's notes (and e.g. those of Bosch) it struck me that the prime spectrum of affinoid algebras never seems to appear, only the maximal spectrum. Can somebody explain the reason for this? 
 A: Another point to bear in mind, in addition to those raised by Brian and Kevin, is that generic points (in the sense of non-maximal prime ideals) don't make sense in analytic geomtery.
For example, the Tate algebra $\mathbb Q_p\langle\langle x\rangle \rangle$ contains 
one non-maximal prime ideal, the zero ideal.  Geometrically it corresponds to the closed 
disk $|x| \leq 1$.  Where in this disk would the generic point corresponding to the zero
ideal live?  The point is that, unlike in algebraic geometry, in rigid analytic geometry one can find disjoint open subsets of irreducible spaces such as the closed disk.
In Berkovich's theory, one does have generic points, but they consist of more data than just a prime ideal; one must also choose a norm on the residue field.  (This relates to Brian's comment.)  Geometrically, this choice of norm pins down where on the rigid space the generic point lives.
A: I am surprised that Brian got to this one first without making what I thought was another obvious comment: affinoids are Jacobson rings! A function which is zero at all points of an affinoid rigid space corresponds to an element of your affinoid algebra which is in all maximal ideals and hence (by Jacobson-ness) is nilpotent. For a general ring this certainly isn't true: the intersection of all prime ideals is the nilpotent elements, but the intersection of all maximal ideals might be bigger (think of a 1-dimensional local ring, for example). 
