The Riemann hypothesis is proved over function fields (like the fraction field of F_q[t]), not finite fields, and the "real version" is a question about the integers.  Kakeya is proved over finite fields, and the "real version" is a question about, well, the reals.  So the situations are quite different.  I'd say that the truth of RH over function fields really does make me feel more confidence that RH is true over number fields, because the analogy between function fields and number fields is in many ways a very close one.  On the other hand, the truth of Kakeya over finite fields does not tell me very much about the truth of the real Kakeya problem.  For one thing, what's true over finite fields -- that a Kakeya set has _measure_ bounded away from 0 -- is totally false over the real numbers!  (See:  Besicovich set.)

One reason (but not the only reason) the Kakeya problem is easier over finite fields is that finite fields only have one scale, while the real numbers have multiple scales.

One reason (but not the only reason) the Riemann Hypothesis is easier over function fields than it is over number fields is that the zeta function in the function field case has an interpretation in terms of the etale cohomology of a variety over a finite field.  In the number field case there is no such interpretation at present, despite the best efforts of the F_un-ologists.