A short note in Group Atlas v2.0 states:
$\mathrm{PSp}(4,7)$ is the smallest simple group whose order is a proper power.
Question: Is there other known finite simple groups whose orders are perfect powers? I have checked all finite simple groups whose order $<10^{10}$ without finding out another example.
In
Newman, Morris; Shanks, Daniel; Williams, H. C., Simple groups of square order and an interesting sequence of primes, Acta Arith. 38, 129-140 (1980). ZBL0365.20025.
it is shown that $\operatorname{PSp}(4,p)$ has order a perfect square if $p$ is a prime that appears in the sequence defined recursively by $a_1=1$, $a_2=7$ and $a_n=6a_{n-1}-a_{n-2}$ (see this entry in OEIS). The authors conjecture that there are infinitely many such primes, and state that they don't know any other examples of finite simple groups of square order.
The first two such primes are 7 and 41, so $\operatorname{PSp}(4,41)$ is another finite simple group whose order is a square. (According to my calculations the order is $81898320^2\sim 6.7\times 10^{15}$.