Logarithmic loss minimization leads to well-behaved probabilistic outputs.
Hinge loss leads to some (not guaranteed) sparsity on the dual, but it doesn't help at probability estimation. Instead, it punishes misclassifications (that's why it's so useful to determine margins): diminishing hinge-loss comes with diminishing across margin misclassifications.
Logarithmic loss ideally leads to better probability estimation at the cost of not actually optimizing for accuracy
Hinge loss ideally leads to better accuracy and some sparsity at the cost of not actually estimating probabilities
In ideal scenarios, each respective method would excel in their domain (accuracy vs probability estimation).
However, due to the No-Free-Lunch Theorem, it is not possible to know, a priori, if the model choice is optimal.