Tilted transverse isotropy (TTI) provides a useful model for the elastic response of a medium containing aligned fractures with a symmetry axis oriented obliquely in the vertical and horizontal coordinate directions. Robust methods for determining the TTI properties of a medium from seismic observations to characterize fractures are sought. Azimuthal differencing of seismic amplitude data produces quantities that are particularly sensitive to TTI properties. Based on the linear slip fracture model, we express the TTI stiffness matrix in terms of the normal and tangential fracture weaknesses. Perturbing stiffness parameters to simulate an interface separating an isotropic medium and a TTI medium, we derive a linearized P-to-P reflection coefficient expression in which the influence of tilt angle and fracture weaknesses separately emerge. We formulate a Bayesian inversion approach in which amplitude differences between seismic data along two azimuths, interpreted in terms of the reflection coefficient approximation, are used to determine fracture weaknesses and tilt angle. Tests with simulated data confirm that the unknown parameter vector involving fracture weakness and tilted fracture weaknesses is stably estimated from seismic data containing a moderate degree of additive Gaussian noise. The inversion approach is applied to a field surface seismic data acquired over a fractured reservoir; from it, interpretable tilted fracture weaknesses, consistent with expected reservoir geology, are obtained. We determine that our inversion approach and the established inversion workflow can produce the properties of systems of tilted fractures stably using azimuthal seismic amplitude differences, which may add important information for characterization of fractured reservoirs.