From a purely applied perspective, my experience is that the difference between these methods is typically not huge, leading to qualitatively the same conclusions (see Table 2 in the vignette you referenced). Therefore, choosing one or the other in practice is often more a matter of taste than due to the actual properties. In the classic parametric statistics literature the quasi-Poisson model is well known due to coverage in the classical "Generalized Linear Models" book by McCullagh & Nelder. However, in econometrics, social or political science, so-called "robust" sandwich standard errors are very common and hence also often used in the presence of (potential) overdispersion.
From a more theoretical perspective, the sandwich approach has the property to guard against more potential departures from the Poisson model while the quasi-Poisson is geared directly towards overdispersion. Hence, the quasi-Poisson (as well as the negative binomial) model just requires estimation of a single additional parameter. In contrast, the sandwich covariance matrix implicitly allows for a different variances for each observation and thus needs to estimate more in a certain sense.
