Cosserat elasticity has a characteristic length. There is a length scale associated with toughness. Classical elasticity has no length scale.
Cosserat elasticity predicts reduction of stress concentration factor around holes, cracks, and other stress raisers.
Classical elasticity does not correctly predict strain fields in heterogeneous solids unless the structure size is negligible. Experimentally, stress concentration factors are lower in heterogeneous solids than values predicted classically.