My knowledge of mathematics is fairly limited.
But I know that Sieve of Erastothenes can be used to find prime numbers in increasing order. While Riemann Hypothesis enforces that distribution of prime numbers is not random.
But I still don't get it, what are mathematicians looking for exactly? Because when we already have a sieve for prime numbers...
Erastothenes' sieve allows us to find primes, but it says nothing about the distribution of primes. This is where Riemann's Zeta function comes in, as it tell us deep insights about this distribution.
For instance, Riemann gave an explicit formula for the number of primes less than $n$ that involves the non-trivial roots $\rho$ of the Riemann $\zeta$-function:
$${\displaystyle \pi (x)=\operatorname {R} (x)-\sum _{\rho }\operatorname {R} (x^{\rho })},$$
with ${\displaystyle \operatorname {R} (x)=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\operatorname {li} (x^{1/n})}$, $μ(n)$ is the Möbius function and $\operatorname{li}(x)$ the logarithmic integral function. In fact,
$${\displaystyle \pi (x)=\operatorname {R} (x)-\sum _{\rho }\operatorname {R} (x^{\rho })-{\frac {1}{\ln {x}}}+{\frac {1}{\pi }}\arctan {\frac {\pi }{\ln {x}}}}.$$
Note that this is an equality, i.e., the exact number of number below $n$. Once we find out whether Riemann's Hypothesis is true, in other words that the $\rho$ are all of the form $\frac12 + bi$, with $b\in\mathbb{R}$, we will discover how the primes behave exactly. As it turns out, this is not trivial at all.
On the other hand, Erastothenes' sieve is more of an algorithm, instead of a deep insight into the nature of the building blocks of the numbers.
If you would like to learn more, a quick search on this website will yield many interesting (and much more concise) answers than mine. I highly recommend you do that.