Most resources on proper scoring rules mention a number of different scoring rules like log-loss, Brier score or spherical scoring. However, they often don't give much guidance on the differences between them. (Exhibit A: Wikipedia.)
Picking the model that maximizes the logarithmic score corresponds to picking the maximum-likelihood model, which seems like a good argument for using logarithmic scoring. Are there similar justifications for Brier or spherical scoring, or other scoring rules? Why would someone use one of these rather than logarithmic scoring?