BurstCube will increase the sky coverage for short (<2 s) gamma-ray bursts (GRBs), especially important in the current era of GW discoveries. The recent coincident detection of a short GRB (by Fermi and other multiwavelength partners) and a GW trigger has provided concrete proof that at least some short GRBs are produced by binary neutron star mergers. BurstCube will provide localizations which will assist wide-field follow-up observers in afterglow detection and redshift measurement. This will lead to additional insight into cosmological parameter estimation, constraints on the neutron star equation of state, and an inventory of r-process elements in the Universe constrained by the faint short GRB kilonova signature (seen in the most recent event). BurstCube will detect GRBs (long and short) from the entire unocculted sky providing broadband spectra for bursts detected elsewhere, rough localizations for follow-up, and accurately timed light curves. BurstCube will also detect solar flares, magnetar flares, and other hard X-ray transients, as well as persistent sources via occultation analysis. In a future without other GRB triggering instruments, BurstCube could provide all-sky coverage for a small fraction of the cost of an Explorer.