For a rumour spreading protocol, the spread time is defined as the first time that everyone learns the rumour. We compare the synchronous push&pull rumour spreading protocol with its asynchronous variant, and show that for any $n$-vertex graph and any starting vertex, the ratio between their expected spread times is bounded by $O \left({n}^{1/3}{\log^{2/3} n}\right)$. This improves the $O(\sqrt n)$ upper bound of Giakkoupis, Nazari, and Woelfel (in Proceedings of ACM Symposium on Principles of Distributed Computing, 2016). Our bound is tight up to a factor of $O(\log n)$, as illustrated by the string of diamonds graph. We also show that if for a pair $α,β$ of real numbers, there exists infinitely many graphs for which the two spread times are $n^α$ and $n^β$ in expectation, then $0\leqα\leq 1$ and $α\leq β\leq \frac13 + \frac23 α$; and we show each such pair $α,β$ is achievable.