In the centres of the Milky Way and M83, the global environmental properties thought to control star formation are very similar. However, M83's nuclear star formation rate (SFR), as estimated by synchrotron and H α emission, is an order of magnitude higher than the Milky Way's. To understand the origin of this difference we use ALMA observations of HCN (1 - 0) and HCO+ (1 - 0) to trace the dense gas at the size scale of individual molecular clouds (0.54 arcsec, 12 pc) in the inner ~500 pc of M83, and compare this to gas clouds at similar resolution and galactocentric radius in the Milky Way. We find that both the overall gas distribution and the properties of individual clouds are very similar in the two galaxies, and that a common mechanism may be responsible for instigating star formation in both circumnuclear rings. Given the considerable similarity in gas properties, the most likely explanation for the order of magnitude difference in SFR is time variability, with the Central Molecular Zone (CMZ) currently being at a more quiescent phase of its star formation cycle. We show M83's SFR must have been an order of magnitude higher 5-7 Myr ago. M83's 'starburst' phase was highly localized, both spatially and temporally, greatly increasing the feedback efficiency and ability to drive galactic-scale outflows. This highly dynamic nature of star formation and feedback cycles in galaxy centres means (i) modelling and interpreting observations must avoid averaging over large spatial areas or time-scales, and (ii) understanding the multiscale processes controlling these cycles requires comparing snapshots of a statistical sample of galaxies in different evolutionary stages.