It was proved by Burnside that a permutation group of prime degree must either be solvable, in which case it has a normal Sylow $p$-subgroup and is contained in the affine group, or it is $2$-transitive. See here for a recent straightforward proof of this.

The classification of the finite $2$-transitive groups was completed as an application of the classification of finite simple groups (see the wikipedia page or the discussion here for references) and they are listed for example in Section 7.7 of Dixon and Mortimer's book on permutation groups.

A look through the list shows that the only examples of prime degree are affine groups, $A_p$, $S_p$, groups $G$ with ${\rm PSL}(d,q) \le G \le {\rm P \Gamma L}(d,q)$ when $(q^d-1)/(q-1)$ is prime, and a few small examples (${\rm PSL}(2,7)$ and ${\rm PSL}(2,11)$ of degrees $7$ and $11$, $M_{11}$, and $M_{23}$).

When $(q^d-1)/(q-1)$ is prime $p$, an element of order $p$ in ${\rm PSL}(d,q)$ is a Singer cycle, and its normalizer in ${\rm P \Gamma L}(d,q)$ has order at most $pde$, where $q=r^e$ with $r$ prime, which is smaller than $p(p-1)$.
The small examples are all contained in the alternating groups.

So the affine group of order $p(p-1)$ is maximal in $S_p$ whenever $p>3$.