This is what I know about why someone might care about fields of positive characteristic:
- they are useful for number theory
- in algebraic geometry, a theory of "geometry" can be developed over them, and it's fun to see how this geometry works out
Some people might read this and think, "What more could you need?" But I've never been able to make myself care about number theory, so (1) doesn't help me. (2) is nice for what it is, but I'm hoping there's something more. My understanding of (2) is that this is only geometry in a rather abstract sense and, for instance, there's no generally useful way to directly visually represent these fields or varieties over them the way we can over the reals or complex numbers. (Drawing a curve in R^2 and saying it's the curve over some other field may be helpful for some purposes, but it's not what I'm after here.)
Is there anything else? The ideal (surely impossible) answer for me would be "Yes, such fields are very good models for these common and easy to understand physical systems: A, B, C. Also, we can visualize them and varieties over them quite easily by method D. Finally, here's a bunch of surprising and helpful applications to 500 other areas of mathematics."
UPDATE: to answer Qiaochu's comment about what I do care about.
Let's say I care about:
algebraic & geometric topology
differential geometry & topology
applications to physics
and I certainly care about algebraic geometry over C
(this is to say I understand the motivations behind these subjects and the general idea, not necessarily that I know them in depth)