Why need characteristic p? In elementary number theory, we only need to face the problem under the field $\mathbb{C}$. Everything seems self-consistent. So what is the motivation to discuss number theory under the field whose char $\neq $ 0? 
 A: Number theory, at it's heart, deals with problems over $\Bbb Z$--and of course more general things that come from motivation from $\Bbb Z$. And one of the most beloved subjects is solving quadratic forms. Thanks to Hensel's Lemma and the celebrated Hasse-Minkowski Theorem we know that solving a given integral quadratic form over $\Bbb Z$ is equivalent to solving it over $\Bbb F_p$ for all odd $p$ and $\Bbb Z/8$ for $p=2$ and over $\Bbb R$. This alone is an enormous number-theoretic reason which would justify studying $\Bbb F_p$ all by itself.
Even in elementary number theory, we use modular arithmetic, which--in the prime case--is arithmetic over $\Bbb F_p$, so I would also dispute the underlying claims you make in your problem statement. Congruences are an enormously important part of study in number theory. Dealing with modular arithmetic integers and Dirichlet characters leads us to a result on $\Bbb Z$ by using $\zeta$-functions to prove Dirichlet's Theoreom on Arithmetic Progressions.
There are many, many more reasons to care, even about problems from elementary theory, but I think these two are some of the most central, the latter is closer to pure questions on modular arithmetic rather than specifically primes, but I think in general it illustrates that rings with non-zero characteristic are both interesting and relevant even in the classical theory.
A: For one thing, any finite field has characteristic $p$, $p$ a prime. These fields have numerous applications in cryptography for instance. In number theory specifically, many problems over the integers can be solved by reducing them modulo some prime number(s), which leads naturally to finite fields.
