Asymptote of the prime gap

Question

A famous conjecture in number theory states that $p_{n+1}-p_n=O((\log p_n)^2)$, where $p_n$ is the $n$-th prime number. However, it is well-know that $p_n \sim n\log n$. So why is the leading conjecture not that $p_{n+1}-p_n \sim n\log n$, since $(n+1)\log(n+1)-n\log n \sim \log n$?