The answer is complex but likely relates to the properties of (near-)resonances in the Solar System, which can stabilise or destabilise orbits at quite a long range.
As noted in James K's answer, perturbations from Jupiter are the main reason for the absence of Saturnian Trojans. While the exact process is complex, one major factor appears to be Laplace's "Great Inequality": the near-5:2 resonance between Jupiter and Saturn. As remarked in de la Barre et al. (1996) (who use the term "Bruin" to describe Saturnian Trojans), this relationship is a major factor in the dynamics:
We numerically integrated various Bruin orbits using different Solar System models to develop a Hamiltonian perturbation theory for low-inclination Bruin orbits. Although only at the beginning stages of development, the theory already identifies three separatrices of Bruin motion due in part to the Great Inequality (GI) between Jupiter and Saturn. These GI separatrices are a major contributor to the unstable region near Saturn's L4 and L5 points.
In contrast, the Martian Trojan points can support orbits with lifetimes comparable to the age of the Solar System. Mars and Jupiter are not in a Great Inequality-like relationship, which may explain why the influence of Jupiter isn't as destructive.