The maximal ideal and Jacobson radical often appear in the Banach algebra theory, but I do not see the prime and nilradical in it.
We can define a prime for a Banach algebra following the ring theory, but few books discuss it. Why? Can Banach algebras contain prime ideals which are not maximal ideals?
Yes, even $C(X)$ (for a generic compact hausdorff space $X$ ) has lots of prime ideals which are not maximal. See here and the references given there. In fact as a ring $C(X)$ is $\infty$-dimensional. It makes more sense to consider closed prime ideals.