The drawback of KL divergence is that: suppose the q(w|.) is the variational distribution and p(w|.) is the posterior distribution we want to use. The KL divergence will penalise q(w) for placing mass where p(w|.) has no or small mass and penalise less for not placing mass where p(w|.) has large massSee another note. The authors constructed a new variational bound which is tighter than the KL bound and more mass covering. Compared to alpha-divergences, its reparameterization gradients have a lower variance. In short, the authors chose a lower bound lies in the general version of evidence lower bound (ELBO) - f-ELBO, that is a biased estimator with smaller variance which induces careful bias-variance trade-off.
Note: it also contains a good review how ELBO can be derived from the marginal distribution of data.