Finding the next prime number is faster within the same range after about $10^8$. Eratosthenes is as fast as $O(n)=n\log(\log(n))$, while the best known algorithm for Riemann zeros is far from trivial and about $O(n)=n^{1+\varepsilon}$ where if you want to reduce $\varepsilon$ you need more space.

The difference is that you can find a very large Riemann zero without knowing any previous zero, while you need to find quite some number of first few primes before any given value, if you want to be 100% sure that you have found a prime number. For example, in the most primitive implementation of Eratosthenes sieve, if you have first $n$ primes you can find primes up to $n^2$, but you still need to list first $n$ primes.

You can, however, use some algorithms that are testing if a number is prime, but then you have to scan some region in order to find a prime number. The best known deterministic is way faster than any known Riemann zeta algorithm, it is known as AKS algorithm and has $O(\log(n)^6)$. Above that you are left with quick, but only probabilistic algorithms, for prime numbers.

Overall prime is a winner. For Riemann zeta you do have a quick estimation that are trivial to calculate alike if all zeros are in the form of $\frac{1}{2} \pm i\gamma_n$, meaning Reimann hypothesis is correct then

$$\gamma_n \approx 2 \pi \frac{n-\frac{11}{8}}{W\left(\frac{n-\frac{11}{8}}{e}\right)}$$

where $W$ is Lambert $W$ function.