It is generally assumed that seismic activity at volcanoes is closely connected to degassing processes. Intuitively, one would therefore expect a good correlation between degassing rates and seismic amplitude. However, both examples and counterexamples of such a correlation exist. In this study on Villarrica volcano (Chile), we pursued a different approach to relate gas flux and volcanic seismicity using 3 months of SO$_2$ flux rate measurements and 12 days of seismic recordings from early 2012. We analyzed the statistical distributions of interevent times between transient seismic waveforms commonly associated with explosions and between peaks in the degassing time series. Both event types showed a periodic recurrence with a mode of 20-25 s and around 1 h for transients and degassing, respectively. The normalized interevent times were fitted by almost identical log-normal distributions. Given the actually very different time scales, this similarity potentially indicates a scale-invariant phenomenon. We could reproduce these empirical findings by modelling the occurrence of transients as a renewal process from which the degassing events were derived recursively with increasing probability since the previous degassing event. In this model, the seismic transients could be either produced by degassing processes within the conduit or by gas release at the lava lake surface while the longer intervals of the degassing events may be explained by accumulation of gas either in the magma column or in the juvenile gas plume. Additionally, we analyzed volcano-tectonic events, which behaved very differently from the transients. They showed the clustered occurrence of tectonic earthquakes.